law conversion course?

It works by doing the PGDL (Postgraduate Diploma in Law) after you finish your undergraduate degree. The proportion of law students to non law students who become solicitors and barristers has been around 50/50 for a while now, so there is no advantage or disadvantage either way. Both undergraduate law degrees and the PGDL are broad degrees, and you'll study a wide range of modules in both of them. Fewer in the PGDL because it's shorter, but the subjects are still broad. You'll find that a lot of barristers and solicitors specialise in something that they studied, but that again is chance and not causative. I studied employment law on my law degree and now practise in (amongst another things) employment law, but the fact of me taking that module in my undergraduate degree had no effect whatsoever either on my choice to move into employment law, or my ability to do so.
The best advice with this choice is to do what you are interested in, and what you are likely to get the best grades in. If you are genuinely interested in a different degree, do that degree. You're not likely to get another chance to pursue an academic interest like this, so take that opportunity now. Then convert to law. If you don't, do a law degree. Whilst these decisions don't affect your career prospects in and of themselves, you should see the university process not just as a box ticking exercise (though there are some boxes you need to tick, such as getting a 2:1 or preferable a First at undergraduate), but also as a genuine opportunity to explore different subjects and build your experience as an individual. The effect of that isn't necessarily tangible, but the more you seek to simply improve yourself and gather experiences, the better chance you have of being a stronger candidate when you come to applying for training contracts or pupillage.