Sorry for the questions first...

Do you want the therapy, or is it your friend's idea?

Why can the public/community mental health services not meet your needs? Those programs get new therapists and there might be someone there now who wasn't there before, and who could be helpful.

I would first make lots of calls before deciding that private therapy isn't affordable.
My therapist has reduced her usual fee for me and I pay out of pocket less that some might pay with an insurance copay. Even with that, I had to reduce my expenses and give up cable, etc. To me, therapy is a necessity as much as food and shelter.

I would accept the friend's help and ask what the time limit so you both know what that is. Perhaps you can barter if there is something you can do for your friend.
Perhaps difficulty accepting help from your friend reflects an issue that therapy could help, if that help could be accepted.