OP, I am sure that you can recognise marketing when you see it. Large numbers of people pass through UoL and BPP in order to obtain a PGDL and/or take professional exams. Some of those people (by no means all) obtain jobs with law firms or become barristers. There is no correspondence between success and the postgraduate law school attended. You can form a view about UoL by the non-stellar A level grades which it asks of those who take the LLB there.
Despite claims to the contrary, no law school is any good at teaching the practical aspects of lawyering. You can only learn to be a lawyer by lawyering, just as you can only learn how to fly by flying, how to ski by skiing, how to cook by cooking, and so on.
I can't speak for all the law firms, but my experience in chambers is that we aren't much interested in where an applicant obtained a PGDL or took the Bar exams. We are interested in how well the applicant did at university, whether studying law or not, as we see that as a more meaningful measure of ability. We are quite interested in people having LLMs or BCLs, but only if they have good first degrees, not an LLM obtained as degree-washing.
We don't show the selection panel the names of the applicants' universities, but it tends to be the case that candidates who have studied at the most competitive universities do well in selection exercises. My chambers' two most recent recruits have, respectively, a first in Modern History from Oxford and a PGDL, and a 2.1 in Law from the Open University plus experience in an NGO.
I would hope that the KCL course which you are looking at doesn't operate inside a silo, not connected to the rest of KCL's academically reputable law school. You might obtain a better quality of legal education at KCL, and that could be useful later on. Looking back at my Diploma in Law, I was well taught in some subjects and not in others. I did a lot of self-instruction, and once I was in practice taught myself other subjects, such as company law and conflict of laws. Bar school was not very useful, but the exam had to be passed. The course has changed since then, but still doesn't seem very good.
Money may be an issue. If one of the courses you are looking at is a lot cheaper than the others, that might be a decider, but I suppose that the prices may be much of a muchness.