The Lex Puteoli Inscription (first century BC) is somewhat ambiguous because it is unclear whether it describes patibulum-bearing or workers bringing patibula to the execution site: "Whoever will want to exact punishment on a male slave or female slave at private expense, as he who wants the punishment to be inflected, he exacts the punishment in this manner: If he wants to bring the patibulum to the cross (in crucem patibul[um] agere), the contractor will have to provide wooden beams (asseres), chains, and cords for the floggers and the floggers themselves. And anyone who will want to exact punishment will have to give four sesterces for each of the workers who bring the patibulum (patibul[um] ferunt) and for the floggers and also for the executioner. Whenever a magistrate exacts punishment at public expense, so shall he decree; and whenever it will have been ordered to be ready to carry out the punishment, the contractor will have gratis to set up crosses (cruces statuere), and will have gratis to provide nails, pitch, wax, candles, and those things which are essential for such matters" (II.8-12). As you point out, this may simply be a matter of workers bringing the patibulum along with other materials to set up the execution apparatus, in which case it wouldn't refer to patibulum-bearing. Even if this is the case, this is still a matter of the patibulum being brought to the crux, which is itself set up (statuere) at the execution site, so it is clear here that patibulum is not synonymous with crux. But John Cook (NT, 2008) makes a pretty convincing case that the inscription refers to patibulum-bearing by the victim. The verb agere, which is loosely translated "bring" by Cook, has more of a sense of "impel, push", which is intelligible in the case of forced patibulum-bearing involving floggers (indeed, it is the usual word for referring to the driving of animals under a harness or yoke). The floggers may thus have been the workers who move or impel the patibulum to the cross by flogging the slave carrying it. Since patibul[um] is incomplete in the text, it is also possible that the word was patibulatum and the sense is "If he wants to impel the person attached to the patibulum to the cross". The phrase in crucem agere, in fact, occurs elsewhere, where it pertains to the person condemned to the cross: "You dared to lead someone off to the cross (in crucem tu agere ausus es)" (Cicero, In Verram 2.5.163), "He was led off to the cross (in crucem ageretur)" (Cicero, In Verram 2.5.165), "The student is led off to the cross (agitur paedagogus in crucem)" (Calpurnius Flaccus, Declamationum 23), "A prostitute leads off to the cross her slave who is in love with her (meretrix servum suum amantem se in crucem agit)" (Calpurnius Flaccus, Declamationum 33), etc.
In this inscription, the term "crucem patibulum" is an asyndetic group, ordinary construction in epigraphy, which "peut sembler faire difficulté" (may seem to challenge) as indicated by F. Hinard and J. C. Dumont (supra, p. 117) in their book.
In their translation, they are choosing to introduce into this expression the conjunction "and" translating the phrase "si in cruc(em) patibul(...) agere uolet" by "s'il commande une exécution par la croix et le patibulum" (if command execution by the cross and patibulum). Two instruments are associated here, sending each to a punishment.
Translators J. F. Gardiner and T. Wiedemann introduce, for their part, the conjunction "or" and translate the passage thus: "If he wants to put the slave on the cross or fork". Here, the master can choose between two instruments, two possible punishments (crux or patibulum). Although the meaning of the phrase is different, even opposite, neither is an incorrect translation, the asyndetic group requires, indeed, restore a conjunction to make the text clearer.
G. Cook gives yet another translation: "If he wants to bring the patibulum to the cross". His translation draws attention to the fact "to bring the patibulum to the cross". Cook goes beyond the Latin text. He is in the interpretation. Hinard et Dumont see in this passage as a possible reference to this custom, going beyond their translation of the Lex Puteoli, but they are openly attempting to interpret the text. If this translation of Cook may agree with the context close, without much difficulty, it does not argue for the belief that the wearing of "patibulum" is a systematic prior to the crucifixion. Why?
The Lex Puteoli concerns the situation where a master wants to torment his slave: "Whoever Will want to exact punishment (" supplic(ium) ") were male slave or female slave at private Expense". Lines 8 and 12 show that the punishment is the crux.
The text specifies the procedure: "as he who wants the punishment to be inflected, he exacts the punishment in this manner (ita) : If he wants to bring the patibulum to the cross". The conjunction "if" (lat. si) suggests the possibility. However, the entire text does not appear that to use a "crux" as a possibility: "the contractor will have gratis to set up crosses". Perhaps only the "patibulum" is optional. It is, moreover, suggest the sentence Cook: "If he wants to bring the patibulum..."
This law regulates the tortures in two settings: private and public. Note that the execution is carried out as public is clearly on a crux without any mention of patibulum, unlike the execution is carried out in private: "Whenever a magistrate exacts punishment at public expense, so shall he decree; and whenever it will have been ordered to be ready to carry out the punishment, the contractor will have gratis to set up crosses, and will have gratis to provide nails, pitch, wax, candles, and those things which are essential for such matters".
It may also question whether two different punishments (see supplies each time).