Have you tried talking about this with your manager? Jumping straight to FMLA is making a lot of assumptions about the company and what they might be willing to do to retain you.
As an example: my SO was dealing with a lot of stress due to health reasons and tried to use FMLA. Her company let her do the paperwork, but ultimately let her take off 3 weeks of paid leave that didn’t count against her PTO. That gave her some room to breathe, but she started crashing again a few months later and tried to hand in her resignation. Welp. Her company is now giving her 6 months of paid leave with no hard obligation to return to work, just a request to be courteous and let them know if she’s not coming back so they can stop paying her.
If the first time that your manager is hearing about your situation is also your request to start taking FMLA immediately or else resign, your manager isn't going to be able to do much for you (and probably won't want to either, given the tough situation you will be putting them in on short notice.)
IMO our industry should really start considering sabbaticals as the next standardized perk. I think they would do a lot to prevent burnout and retain top performers.