No-Name IB -> MBA -> BB/EB? From London to US?

Hi all, currently a final year undergrad at a target in London and returning full-time at a LMM IB boutique in London having done a stint in the summer. I am looking to ultimately move up to the BB/EB/MM banks in most part due to modelling experience and excitement of doing large deals (I want to be a career banker).

Given my circumstance (and assuming I don’t pursue any other opportunities), my plan is as follows:

1. Work for 3 years and make Associate at this no-name IB boutique
2. Kill GMAT
3. Apply for M7 MBA programmes in the US (having a very highly ranked undergrad I hope will help)
4. Recruit for BB/EB (Summer) Aso roles during my MBA in the US
5. Stay in US (London hosts less and less structured post-MBA programmes every year)

I have honestly never considered this route and am completely new to this game. How does this plan sound? Is it viable at all and if not why?

Thanks guys.