Narkissos....Indeed, this verse is a notorious crux interpretum! Perhaps its problems are due in part to the fact that it is an adapted second-hand text and the author of James didn't take the trouble to smooth over the language to fit its new context. This means, as you noted, that what the author of James meant and what the passage meant in its original context are two separate issues. The translation of the verse that I employed, btw, is that of Dibelius who gave a pretty thorough analysis of the verse in the commentary.
As for taking pros phthonon as adverbial, Dibelius cites parallel expressions like pros orgén "wrathfully" or pros alétheian "truthfully" in the literature: "...and ceased not to pull me down violently (pros bian)" (Philo, Spec. leg. 3.3), and "Do not look at another woman lustfully (pros epithumian)" (Clement of Alexandria, Strom. 7.82.3). I agree that a sense of opposition is possible but if the adverbial use was common with emotive nouns like epithumias "lust" or orgés "wrath", the oppositive sense for pros + phthonon "envy" may be less likely. Bede proposed your solution as one possibility of several: "Its desires are against jealousy (adversus invidiam concupiscit) that the sickness of jealousy be vanquished and rooted from your minds" (Super Divi Jacobi Epistolam, MPG 93, 9-42), but Dibelius feels it is inferior to the adverbial sense. It would be good to check what other critical commentaries might say.
As for what the subject of epipothei "he yearns" is supposed to be, this may be a case where James and the original context of the citation parted company. It may well be that the author of James intended one reading while the original source assumed another. It's hard to tell what to make of all the different clues. If we take pneuma as a logical object (as given in the translation by Dibelius which takes the verb as pro-drop relying on a prior antecedent outside the quotation for interpretation), this fits well with the presumed Eldad and Modad source; the Spirit was what was being envied, someone without the Spirit was "jealous" of others having it (Numbers 11:29). The author of 1 Peter also appears to have interpreted the pneuma as the object of epipothei. 1 Peter which is strongly dependent on James (to an extent similar to Ephesians' dependence on Colossians), and the beginning of ch. 2 utilizes James 4:5 and the surrounding context in its admonitions....perhaps it is thus the earliest interpretative use of the difficult passage in James:
"Therefore, rid yourselves of all malice and all deceit, hypocrisy, envy (phthonous), and slander of every kind. Like newborn babies, crave (epipothésate) pure spiritual (logikon; cf. Romans 12) milk, so that by it you may grow up in your salvation, now that you have tasted that the Lord is good....I urge you, as aliens and strangers in the world, to abstain from sinful desires which war against your soul (epithumión haitines strateuontai kata tés psukhés)" (1 Peter 2:1-3, 11).
"What causes fights and quarrels among you? Don't they come from your desires that battle within you (hédonón ... strateuomenón en tois melesin humón)? You want something but don't get it. You kill and covet (zéloute), but you cannot have what you want. You quarrel and fight. You do not have, because you do not ask God. When you ask, you do not receive, because you ask with wrong motives, that you may spend what you get on your pleasures. You adulterers, don't you know that friendship with the world is hatred toward God. Anyone who chooses to be a friend of the world becomes an enemy of God. Do you suppose it is in vain that the Scripture says, 'He yearns (epipothei) jealously (pros phthonon) after the Spirit (pneuma) which he has made to dwell in us?' " (James 4:1-5).
1 Peter uses two words from the "Scripture" cited in James 4:5 (phthonos and epipothei) and the use is secondary since its author was not directly citing the underlying text and has worked the terms into his own language. The yearning in 1 Peter is for the "spiritual milk" which is likened to the "Lord" (who is being tasted), and this compares well with an interpretation of James 4:5 that makes the "Spirit" the object of epipothei. However phthonos is a negative trait to be abstained from and is not linked to the "yearning," unless "yearning" for the Lord is understood as the antithesis of sinful envy. As suggestive as this may be, we are only dealing here with a later writer's interpretation and creative use of the passage.
If James and his scriptural source were congruent with respect to the object of epipothei being pneuma, I suspect they could have parted company with respect to the what the subject of the verb was supposed to be. This is because if pneuma is not the subject, we have here a pro-drop construction in which the logical subject would lie outside the quoted text and within the matrix provided by the author. This subject would have to be "God" (the antecedent being theou in v. 4) and not the fuller expression "enemy of God" because the following two verbs in v. 4-5 require "God" to be their logical subject. This concept of divine "longing" is itself not impossible for Theodotus has epipothei as the verb in Job 14:15, where it applies to God ("You will long for the creature your hands have made"). The appelation "adulterers" in v. 4 (which, btw, is feminine in gender, which fits well with the concept in Numbers 5) also could invoke the marital "longing" on the part of God as jealous "husband". Compare this statement in the Pseudo-Clementines:
"For every human being is a bride (numphé) when, after the bright word of truth is implanted by the True Prophet, the mind is illuminated. Therefore, it is necessary to heed solely the one True Prophet, knowing that whoever is implanted with the word of another is charged with adultery (moikheias) and is cast out by the bridegroom from his kingdom" (Hom. 3.27.3.-28.1).
I wonder if the adultery analogy in James works in this way: (1) God bestows his Spirit on those who are faithful to him but (2) withdraws his Spirit from those who are unfaithful. In Hermas, the "Holy Spirit ... departs from a person with the evil spirits," the evil spirits being negative emotions like wrath, temper, and vengefulness (Mand. 5.4-6), and when the Spirit has left the person he is "empty (kenos) of the righteous Spirit" (Mand. 5.2.7). Meanwhile, Hermas spoke of the "angel of wickedness" who "enters your heart (epi tén kardian sou anabé)" (Mand. 6.2.4), and went on to refer to the Devil dwelling inside a person like the Lord: "The Lord dwells (katoikei) in patience, but the Devil lives in an angry temper. So if both spirits live together, it is unfortunate and evil for that person in whom they dwell" (Mand. 5.1.3-4). I find it striking how this statement has the Lord himself "dwelling" in the follower, or the Devil "dwelling" in the unfaithful one. Should we be thinking of this in a quasi-sexual sense, since unfaithfulness is characterized in James as "adultery"? It is striking, in view of Hermas' statement of the Devil dwelling in the unfaithful one, that James says: "Submit yourselves therefore to God. Resist the Devil and he will flee from you" (v. 7). Hermas also says: "If you resist him, he will be conquered and flee from you in shame" (Mand. 12.5.2; cf. 12.4.7). Incidentally, I just found further evidence that a pseudepigraphon may lie behind James 4:5-8, for similar phrases to "Draw near to God" and "The Devil will flee from you" occur frequently in the Testaments of the Twelve Patriarchs:
"You do these as well, my children, and every spirit of Beliar will flee from you ... so long as you have the God of heaven with you and walk with all mankind in purity of heart" (Testament of Issachar 7:7).
"If you achieve the good, my children, men and angels will bless you ... The Devil will flee from you, wild animals will be afraid of you" (Testment of Naphtali 8:4).
This one is amazingly close to James and Hermas:
"The spirit of jealousy and pretentiousness kept saying to me, 'You too are his son.' And one of the spirits of Beliar was at work within me....Anger is evil, my children, for it becomes the motivating force of the soul itself....When the soul is continually preturbed, the Lord withdraws from it and Beliar rules it. Observe the Lord's commandments, then, my children, and keep his Law. Avoid wrath, and hate lying, in order that the Lord may dwell with you, and Beliar may flee from you...Throughout all your life love the Lord, and one another with a true heart...Draw near to God and to the angel who intercedes for you" (Testament of Dan 1:6-7, 3:1, 4:7-5:1, 3, 6:2).
Cool, huh? Here is another one from the Testament of Benjamin:
"Even if the spirits of Beliar seek to derange you with all sorts of wicked oppresion, they will not dominate you....The deliberations of the good man are not in the control of the deceitful spirit, Beliar, for the angel of peace guides his life. For he does not look with passionate longing at corruptible things, nor does he accumulate weath out of love of pleasure. The good set of mind does not receive glory or dishonor from men, nor does it know deceit, or lying, or conflict, or abuse. For the Lord dwells in him, illuminates his life (exactly what the Ps.-Clementine Homilies says!), and he rejoices in everythng at every appropriate time. The good set of mind does not talk from both sides of his mouth ... there is no duplicity in its perception or its hearing. (i.e. not double-minded)" (Testament of Benjamin 6:1-6).
This last one reflects James' concern about materialism and wealth and his "double-minded" statements. Anyway, I like your suggestion of translating James 4:5 as "He/God desires the spirit he has made to dwell in us rather than envy," as this reflects well the contrast between the "spirit of envy" and the "spirit of righteousness", but again I'm not sure whether pros can be taken to be oppositional here. I also consider the question open as to whether the underlying text for James 4:5 might have a different presumed subject for epipothei. On the other hand, having the "spirit" as the subject could correspond to such texts as Psalm 41:2 (LXX) ("epipothei hé psuché mou pros se").