It really depends on what you want to do.
Anything can be comped together, you could render the fx out of max and the rest of your scene out of Maya and comp them in AE or Nuke (Nuke is better but more complex). I think this approach is fine but relies heavily on your confidence in your coming abilities, and getting your render files set up correctly. If you don't know how to set up a comp this complex, maybe bail on this approach. Or check out some tutorials, there are so many out there.
The other option, if you feel less certain about comping, would be to bring everything into one program and render it all at the same time. This has a lot of benefits! You will be able to see your full scene and manipulate it at the same time. This way will also let you separate your scene at render time to get comping layers without the guesswork. But the downside is you'd have to transfer between Max and Maya, and while that's totally possible. You would need to figure out a transfer system that works best for your workflow. USD is great but very complex, Alembic is simple and robust (but you may have to rebuild shading networks). Render proxies are great if you are using the same engine in both programs.
All I'm saying is do what you feel comfortable with, and what you know you can pull off without too many issues. I hope this helps in some way, let me know if you got more questions. good luck!
