1yr M&A to FoFs/Growth Equity

I'm currently working in a balance sheet bank (think of Nomura BNP SocGen HSBC) in London. I was unfortunately sent to a sector I'm not interested at all and there's very few deals in the past 3 yrs.

At the same time, I'm sure that banking is not my thing, based on the past experiences, I wanna exit to Growth Equity and FoFs.

Graduated from a top-tier bs school, didn't do banking before, had 3-4 off-cycle experiences in a in MF and UMM funds and FoFs but didn't get return there as they only take lateral hire for 2-3 yrs.
(Not sure if these are helpful for lateral)

Understand my bank brand is weaker than BB and EB,and I'm in a shitty team with no hope for deals

Can my previous offcycles and school compensate for a weaker IB brand and limited deal experience when recruiting for growth equity or FoFs?

Or should lateral to a BB/EB first to improve my exit opportunities?

Any advice will be much appreciated!