Networking into an Elite Boutique as an Experienced Hire

I've seen a lot of posts about people finding a way into a regional boutique, BB or MM banking role, but I wanted to know if this was doable at elite boutiques given their small size and BB-caliber competitiveness? Is 3 years prior experience too much for an analyst role?

Also, if one was looking to do PE down the line, how would one be viewed by headhunters / PE funds if he had a few years of prior experience (3 years before I enter banking)?

Finally, if it is possible to join, is July/June my ideal start target if I start networking now? It seems like a lot of people get their bonuses and leave about then, but it's also when new hires start. And, would I have access to training as an off-cycle hire?

Background: Second year analyst at an early stage VC (non top-tier), prior to that spent a year doing engineering/medical research (data analysis and programming). HYP engineering undergrad. VC has been great! But, I've realized my universe of options is confined to other early stage funds and operating roles at smaller companies. I want to make this move at this stage in my career to open my doors to working (as an investor or employee) with larger companies.