It is difficult, but not impossible. I think you do need the first. I am not sure you need anything else. If you did the BCL that would help. Few other postgraduate degrees are worth it from the perspective of obtaining a pupillage.
Nowhere near 'most' barristers are from Oxbridge and I am getting a little weary of pointing this out. If people actually did some research (try looking people up on Chambers' websites) instead of mouthing off it would say more about their own abilities.
I agree that most Oxbridge 2:1s will get an interview. That is because standards are high there and most will have had sustained academic success. I also think most 1sts, from anywhere, will also get an interview. Mini-pupillages at the place to which you are applying, which have gone well, will achieve the same thing.
Finally, I wouldn't be so quick to attribute failure to 'snobbery at the bar' if I were you. In most places it doesn't exist, but that isn't the main reason. Cultivating the habit of putting in place an excuse for failure which involves blaming other people is neither attractive, nor desirable in a barrister. The first question I would ask if you applied is why you ended up at an ex-poly. There are some good reasons, but the most likely explanation is that you failed to do well enough in your A-levels. In other words, if you fail to get a pupillage it may - shock horror - be down to you.