As you said, he did that for money. You'll need to find another source and eventually you'll have no big choice : eventually the selling of tax offices like in the IV seems unavoidable and having it without citizenship...So what if this edict is never issued?
You have another motives though for this edit : as said, you had in the III century the struggle between senatorial, roman elites and military provincial ones. Making such edict was probably a move towards provinces in hope to have them supporting the emperor.
Finally a simplification of the roman law (that was the Holy Grail of many emperors, as Theodose or Justinian) is probably to be noticed.
For the barbarisation of Roman army...It played a role but not directly : indeed it's more a late III century feature coming from the need of replacing auxiliaries but critically to answer the germanic threat more effitiently by using barbarians letae.
Futhermore, many germans were already used as auxiliares as far than the Ist century BC. So, blaiming Caracalla edict for the "barbarisation" of the roman army is a bit short-sighted.
Probably that, at least, the edict forced the army to reorganise itself...Or at least would have allowed without the Military Anarchy.
The High Imperial roman legion was simply too slow, not fitting the new needs of the western front.
A note about women and the edict : I searched in all the text and I didn't found one occurence of this. It's really astonishing anyway, being in total contradiction with all roman tradition and laws.
So back to the question, I would see this, not mutually contradictory.
1) Edict is made by another emperor in need of money. Maybe it would be made by separate ones but finally you'll have it.
2) Only part of the edict is made ATL, but the army ends by being an even more conservative body than OTL and maybe lost there an opportunity to adapt itself.
3) As the taxes remains more in the local institutions rather than imperial ones, you'll have a more important rise of localism and regionalism. More odds of having "secessions" in the empire during the crisis.