The actual university you attend is largely, if not completely irrelevant now when it comes to pupillage applications. It was never really as big a factor as applicants thought it was, but many sets are now anonymising university names for the paper sift anyway, and for those sets it is literally irrelevant. The reason why more students from better universities get pupillage is because better universities tend to take better students. Better universities may, of course, have better facilities and better standards of teaching, but they also do take the 'best' students. Students from lesser universities can and do secure pupillage, but statistically it is simply the case that if you don't do as well in your A-Levels as other students, you are less likely to have the ability required to become a barrister.
In your case, the other issue is that you have achieved A-Levels that now put you one step behind everyone else. In the grand scheme of things those A-Levels are not going to prevent you from getting pupillage. Your application will not be rejected solely due to your A-Level results. But you will be competing against other driven and able applicants who have all achieved much better A-Level results, so the onus is on you to show that your A-Level results do not reflect your ability, and to achieve at a significantly higher level from this point on. That is along with doing the other extra curricular activities that one would expect from a strong pupillage applicant. It's certainly not impossible, but you have made a less than ideal start, and you do need to be realistic about whether pupillage is a genuinely achievable aspiration for you. You don't necessarily need to decide that now, but it is something you should decide before doing the Bar course. I agree that you should certainly have a Plan B in case it turns out that it is not realistic.