The way it worked for me is:
I enrolled in college (aviation major) while getting my PPL out of pocket. Once I had my private ticket, I was able to start instrument the following semester, with the flight school that the college is partnered with.
The GI bill provided the funding upfront for 50hrs flight + 10hrs simulator(we had a redbird FMS), so I had no monthly limits etc. Being cut off at a grand per month would take me half a century to finish my IR.
That being said, once that funding ran out, I had to start coming out of pocket.
I'd start looking around at colleges and see if any have aviation programs instead of just 141 schools. That way you can get a degree out of it as well as a monthly housing allowance(on or off campus) if you go at least half-time.
Many may even offer different tracks. For instance, I have the option of taking the instructor path and pursuing CFI/CFII, or going for commercial UAV training instead.