I will be completely honest here and say that it seems to me that you may be the type of person that tends to be pushed/manipulated and controlled by others and are minimizing this in the hopes of not "bashing" your family that you feel you are obligated to give a pass because they help you out so much financially.

My point being how you had to emphasize that they are "not controlling" and the financial help is "no strings attached" before pointing out the real reason for this post which is you being pressured to get another opinion on your mental health care.

"despite my improvement..." so in other words they disregard what you tell them has been progress in your mental health. So they ignore that you say you've improved and although you minimize it, it seems to me they are pressuring you to do something that would normally not be anything they have a say in... but then, they support you financially and honestly they feel they have a right to tell you where to go for help. That, to me seems to be the brutal truth even if you minimize or avoid it.

If it weren't for the financial support, you'd be able to entirely just say "no thank you, I am working on it and making progress" But the reason you are here is because you feel obligated somehow or compelled to give in to their demands (or suggestions as you called it)

"..Despite them finally understanding me and my condition..." This implies that you've taken some time to actually have to convince them of your condition.

I think the words you've used here and how you've described things speak volumes to what seems to be going on here.

Can you get support financially elsewhere? First thing I would do is seek other support in this area so as not to feel obligated in any way to comply with their desires.