Whether the wage increase is contractually required to be retroactive to the first of the month depends on the exact wording of the agreement, and I mean the entire agreement, not just a sentence or two pulled out of context. We can't tell from here if it's actually a contract or not and if so, what the requirements are.
Sometimes, it makes sense to stand on your rights and demand what is yours, and other times it makes sense to accept that life isn't always fair and you don't always get everything you are due. Legally, this is your wife's battle and not yours, and we are talking about the difference between her old wage and her new wage for a maximum of two weeks. If SHE (not you) feels that this is the hill SHE (not you) wants to die on, then she needs to take the entire agreement to a local attorney for review. If SHE does not think so, then you need to back off and stay out of it.

