You have excellent GCSE results, so there is no reason you shouldn't be able to apply to LSE, UCL or any other top course (but do include at least one or two courses that are slightly less oversubscribed just in case when you apply).
Both are suitable for continuing on to become a solicitor or barrister, however consider the financial implications of the latter option, as it takes longer to get to that point and there is less funding in general (specifically, there is no governmental funding for GDL/conversion courses, so you would be applying for a training contract position where they will also cover the costs of the GDL; these are very heavily contested, so the former route is statistically and financially a better option).
But in general, no, there is no reason to not do the conversion route (which is only, after all, a year longer typically than doing a qualifying law degree as your undergraduate course) if you really would like to pursue that. Some joint honours courses are QLDs. although you'll need to carefully investigate to see if there are any with history which fulfill this criteria (if you'd be interested in studying both and not having to do the conversion course potentially). Message their admissions team for the law course if you're unsure!