I'm guessing this will be very difficult unless you make compromises. Perhaps lots of compromises.
Certainly you can change institutions, provided you get accepted. That is the least of the problems, I'd guess.
Taking your research with you rather than starting over, however, depends on finding a supervisor at the new institution who is willing to advise you on this topic. That might be harder, depending on how "mainstream" your research is.
Taking the funding along may be harder still. It depends on lots of things, including the willingness of the current institution (since they charge overhead against most grants) and the funding agency. This might be the hardest part of all.
You might, however, be able to get some alternative funding, depending on your field.
But, also consider that you can work on mental health issues through a professional, perhaps at your current institution. That should be a first priority in any case. If you can manage that, the simplest option might just be to stay and finish.
But, depending on the mental health issue, starting over might not be the worst option.