small tech boutique to Tier1/2 Tech PE

I am writing to ask for some advice on making the jump from a small boutique. Any tips would be greatly appreciated.

Current situation - Yr 1 Assoc at a tech boutique in London. I have had experience (long term internship) in a European Investment bank (Tier 2) and degrees from target Universities. I had to restart in a small bank, as I had a 1yr+ gap due to medical reasons. I am 28.

Where I want to be in next 5 years - A decent Tech focused PE or Corp Dev in a Tech company

Question - Do I need to have a stint in a BB or somewhere like Jefferies, Rothschilds to break into Tech PE / Growth Cap (e.g. Mubadala, Summit, Insight, Battery, Francisco Partners, Bridgepoint, etc) or is it better to try post MBA. So far from my experience its been hard  trying to move to one of those mentioned banks, and I would certainly have to drop to Analyst level, which I am hesitant to do so given the age. I am not sure what to do. Is it better to do MBA and get the network or go back to Analyst level and spend 2 years in a bigger bank instead?

Many thanks in anticipation!

Kind Regards