The staff at Kings / LSE and Birkbeck all know each other, if you do well on any of the MSc courses and are a good prospect for a PhD, it won't matter which masters you have done. It is more important to choose the course that you will do really well at. i.e. if you are good at probability/analysis Kings is better than LSE, while if you are good at stats, the LSE is better than Kings.

I did an MSc at Kings and was then invited to do a PhD there in Fin Maths.