"SI IN CRUC PATIB AGERE VOLET" -- Lex Puteoli
I admit, it is a difficult walnut to crack, even though the interpretation of "CRUC" is obvious: "crucem.*" "PATIB" on the other hand, could be interpreted as "patibulum" (noun - direct object - accusative case), "patibulo" (noun - indirect object or instrument - dative or ablative case), "patibulatum" (adjective direct object - accusative case, singular). I myself am partial to "patibulo" and "patibulatum" because to me, the interpretation would make sense in light of the rest of the punishment in lines 8-12. So, then, we would have the options:
si in crucem patibulo agere volet = if he wishes to impel [the slave] onto the crux on / from / with a patibulum.
si in crucem patibulatum agere volet = if he wishes to impel the patibulated (yoked) [slave] to / onto the crux.
In each case it involves assuming that the slave is intended as the direct object.
Bringing the patibulum to the place of execution, on the other hand, appears to be using an incorrect verb -- in the Lewis and Short entry "ago", it appears a better choice of a verb would have been "ferre." (Lewis & Short "ago", I. Lit., C.) Plus, it is inconsistent with the later requirements that the client pay for the floggers and the scourges not to mention the blokes ferrying the patibulum, and the patibulum itself. So, yes, Cook is interpreting beyond the text.
Then there is the use of the preposition "in" used with the accusative. "Agere" is a verb of motion, but it is also a verb of a different kind of forcing, i.e., penetration. (See Augustan Histories, Elagabulus 5.1 -- less rude examples of forcing or penetration can be found in the above L&S entry "ago".) Using it with "in" therefore would be a pregnant construction, "i.e., to bring into... and place there." (See Lewis & Short entry, "in", II. with acc., A) If it was meant as impel to the site of execution only, I think "ad crucem" would be a better choice of words because that means with the accusative a sense of approach. (See Lewis & Short entry "ad", I, A, (a)) Therefore I interpret "in crucem agere" as impel to the object of execution, and force the condemned onto it.
Plus, I have found an appalling number of cases where the Greek verb anaskolopizow (impale, fix on a pole) and anastaurow (impale, suspend on a pole, crucify) have been translated into the Latin as "in crucem agere" by Byzantine, Medieval, Renaissance, Baroque and Enlightenment scholars. But they have precedence from some of the writers of antiquity. That I'll leave for another day.
Needless to say, it is clear from Lex Puteoli that at the time of inscribing "crux" and "patibulum" were two different things, use of the patibulum was optional (si... volet), and the slave was impelled onto a "crux" until his final contact with it.
* Okay, it could also be translated "cruce" which would mean "in cruce agere" = to force on/with a crux but in all the ancient writings I have translated for myself the number of incidences of "in cruce agere" I have found are exactly ZERO - none!