There are therapists and agencies which will see people for reduced fees - obviously, the less you can afford, the fewer options there will be, but it is worth exploring the possibilities. A charity for whom I work offers low-cost counselling with its graduated students while they work up enough hours to complete their qualifications: we have to be a little more careful with assessment, as some clients' presentations aren't suitable for a comparatively inexperienced counsellor, but for many people it can be an opportunity to do some really useful work at little cost. It may be worth finding out if there are any organisations in your area that do something similar - although to be fair most training establishments tend to have their students go to outside agencies for placements rather than providing them internally, the way we do.
Other options - you may already have exhausted these - are various charities (and don't assume there's just one). Sometimes church groups run counselling services, but that doesn't mean the counselling will necessarily be "holy". And it's probably worth looking up on
http://www.itsgoodtotalk.org.uk/therapists to see which therapists offer a sliding scale of fees, in case there's someone near you.
And don't assume you need a psychotherapist - it may simply be that a reasonably experienced non-CBT counsellor is just the job.
As for your concerns about the DWP IAPT thing - yes, I think this is a worrying development, but I feel that it should be possible to navigate it without being compromised. My view would be that, much as people do with these get-you-back-to-work agencies, if you stick to your guns and know your rights, you can probably bollix up the system quite nicely: for a start, you should be entitled to insist on confidentiality and that your client information is not shared
at all with any other agency, let alone DWP. That in itself will probably put the skids under a therapy referral quite nicely, as the system is probably entirely geared up to them being able to browbeat you into agreeing to waive your right to confidentiality.
Another option, if you do end up being forced into seeing one of their therapists is to reiterate your desire for confidentiality to the therapist - again, they should not disclose anything without your specific permission, and if they do, it is likely that you will be able to make a complaint to their professional body...so make sure that you have got the therapist's name, registration number, name of professional body, etc. at the first session. And, of course, if you have any concerns that your desire for confidentiality might not be being respected, you have the option of attending the session, but refusing to share any information - that way, if they then report back to DWP that you are doing this, they have immediately outed themselves as being willing to breach confidentiality. Whichever way it pans out, I'd be really surprised if DWP is going to be able to put something legal in place that obliges you to attend a therapy process wherein your confidentiality is not maintained, so they'd be on a sticky wicket.
If I were in that position, I'd probably try a third strategy - a kind of "scorched earth" approach where I would hit the therapist with every trauma, horror, emotional meltdown and early life happenings I could think of, and let them try to pick the bones out of that. I suspect they'd "refer on" (ie., get rid of you) extremely quickly. Not very therapeutic, but it would get the job done.