Is a lateral necessary for PE recruitment?

Currently an Analyst at a MM Balance Sheet bank in London (think Nomura, SocGen, Santander) working in the Leveraged Finance team which does mainly passive roles on pretty high profile transactions with an occasional Physical role. Also had some exposure to the M&A team which I have the option to switch to full time, however it is a weak platform (nowhere to be seen on league tables).

My goal long term is PE / Growth Equity - I am ok with being at a MM / LMM fund and am also considering moving to my hometown which is a developed market but not London or New York (suppose this makes things easier as I would have the London brand name).

I was contemplating whether my current setup is sufficient for pursuing this goal or if I should seek to join a stronger M&A platform at a bigger name brand bank? I am quite happy with my comp and WLB, and have become comfortable at my current bank however would of course not want to limit my exit opportunities.