I think you might have to work it out with him as analysis of that sort/cost isn't generally done these days; I think in the past the patient and psychoanalyst often did a pact and you just kept paying, even after therapy, kind of as if you had a student loan for school?
Otherwise, I would get a personal loan if you can or take out a "new" credit card (that has checks if he doesn't take credit cards) or something of that sort if you're able (or have someone who will help you like a parent/very good friend)?
I don't know how long you are planning/wishing to see this doctor and whether your insurance pays anything for out of network (mine pays 50% out of network), etc. but I would do the numbers ($275 x 40 sessions/yr. would be $11,000) and see how much you could get from insurance, how much you could contribute, how much you'd have to borrow "ahead" (maybe borrowing $5,000 and "saving" $50 a week of your own money, by the time the $5,000 was used up (18 sessions/weeks at original $275) you would have an additional $900 saved and would be continuing to save during the usage of that so would have 4 sessions, etc.), think of all/multiple possibilities from all angles.