I've been trying to think of all the cheap protein I use. I shop at Sainsbury's mostly, so that's where I've found the things I list:
Fresh sardines or mackerel. Oily fish like this is filling and if used in a light stew or curry with e.g. beans, gives the whole thing a nice flavour;
Frozen lamb chops, lamb shanks or stewing beef. Not veiny or gristly. I cook from frozen in the pressure cooker;
Ham bone or ham ends from the deli counter. These are often sold off very cheaply at the end of the day and are brilliant for soup;
Frozen uncooked prawns when on offer (which is surprisingly often). I stock up because prawns are a king's ransom otherwise;
Turkey-breast fillets or steaks, mince (breast or thigh). It is soooo much cheaper than chicken. You can do anything with it too, as it takes on flavours very well;
Pork belly and ribs. Fatty, but the right cooking can render off a lot of the fat. Again, you can use it in stews where it flavours well;
A 'hypocritical' recommendation because I can't stand the stuff, but squid is cheaper than most fish. Ditto rabbit if you can find a place that stocks it.
And re fat: it does add calories of course, but if that fills you up (as I find it does) and stops you carb-loading later, then overall you will have consumed fewer calories. This is what I have found (11lbs down in just under a month on a new weight loss programme, many lbs to go). Try using coconut oil for cooking (not cheap, but a little goes a long way). I think you will find you are sated better and eat less overall.
Try carbs from veg (carrot, beetroot, parsnips, etc) and complex grains. They take a lot longer to break down, which tends to delay hunger. Brown rice is so much better for that. Brown rice pudding made with reduced fat coconut milk and sprinkled with linseed would be healthy and filling.
Good luck with it all, it sounds very frustrating.