Personally, I would not purchase a home while carrying a debt load.
For the sake, though, of expanding on your thought of this BIL's mother's house. How far is the commute for your wife to University of Houston. Is it also 30 minutes? Will you both have overlap of commute to take one car in if her school, your work is in the same general area? Is public transit an option from suburbs? Have you estimated out above the PITI for the actual cost adjustment for repairs, maintenance, renovations, wear & tear and fuel costs? See how close you get now to that $1000 rental mark for a downtown rental. And I say $1000, because that is what I'd work with on that figure. $1200 may be the rental you 'want' but $1000 would get you what you 'need', so I'd focus on that and add from the $750 PITI that you've estimated for BIL's mother's home.
With that in mind:
One angle that I would address are downtown garage type apartments (which can be really, really nice on the interior! and may get you close to greenspace/parks for dog & baby), 1 bedroom cottages or MIL suites that are on the property of larger historic homes (hugely popular in our region), or if a roomshare arrangement with someone else who is attending grad school with her may help alleviate costs. Maybe a full timer with a part time job that won't be home much? That may kick in 1/3 of the rent? However, I'm not familiar with Houston or it's downtown. Hopefully, someone will chime in with some advice specific to that area. Creative housing would get you down below your estimated rental cost and can still be a great rental in the very same area that you'd be looking in. Sometimes, even coming with some extra perks like a parking space, use of main house's amentities, or even on a better street aesthetically or otherwise.
In the meantime, cruise rental websites for the downtown area specifically, and see what comes up with more unique housing (aside from condos, apartments, and single famly)
With having a baby, one benefit, is that you are not in a position to have to worry (yet) about elementary school. So that gives you great flexibility.