The fact that you've mentioned that you'd like to join a good, reputable firm suggests to me that you want to be a solicitor, in which case I can't really help. My knowledge on solicitor recruitment is pretty basic, and there are plenty of others who are actually solicitors and/or better placed to advise on that.
Just in case, for the Bar A-Levels matter, but not as much as I understand they do for higher tier firms when it comes to training contracts. Some Chambers do set a requirement of AAB and a 2:1, but plenty don't, and even those that do sometimes will disregard that if you have an otherwise outstanding application. It is, on any reading, a weakness to have poor A-Levels when applying for pupillage, but it is not as significant a weakness as getting a 2:2. It is relevant, but by the time you apply for pupillage it will be quite far removed in terms of time from where you are, so your chances of compensating for your A-Levels is higher. I have seen many applications with less than ideal A-Levels that have gotten pupillage interviews, just because there is so much you can do from this point on to make up for them. It does depend to a degree on just how bad they are, but simply put, if your application tells me that you're a strong applicant at the age of 23 or 24, I'm not going to be overly concerned about where you were at when you were 18.
This is not a post advising people with bad A-Levels to disregard them and aim for a career as a barrister anyway. The Bar is still extremely competitive, and everybody should be taking time to make a realistic decision about whether it is worth the risk to aim for it. If you have bad A-Levels, you should taking longer to make that decision, because you're off to a bad start and it's not a process that is forgiving to weaknesses on an application. But, as I say, you may still decide that it is a realistic goal that you want to pursue. And of course, whilst it does come with expense, a law degree is only the next step. You're not having to commit to five or six years of study and pupillage hunting yet.
In the event that you (or someone like you) does want to still go down the pupillage route, my advice as to the options is to go to a bad university and get a First. The university you go to does not matter in the main nearly as much as people think it does when it comes to pupillage, in fact some sets are starting to anonymise the university name on applications. It's all about the degree classification, and with bad A-Levels you really want that to be a First if you want to stand a realistic chance of compensating for the A-Level results.