PE recruiting as a lateral to a top bank?

Just starting as a FT analyst at a top MM this summer. I'm interested in moving to PE, hopefully a MF, and honestly people from my bank just aren't able to make that exit due to the size of the deals we work on and prestige.

I've been thinking about lateraling and wondering how that plays out for PE recruiting. I have some contacts at top banks (both BBs and EBs) and, given the solid reputation of my bank, I think I have a decent shot as a lateral candidate.

Theoretically if I were to lateral to a top bank, would I be on an equal footing with candidates who started at that bank for recruiting? My fear is that, with the way on cycle works, if I lateraled in over the summer (after doing a full year at my bank) I would be 'late to the party' from the perspective of recruiting. I know it's typical that 2nd years send a list of incoming analysts to HHs, so I would need to make sure I was in a position to be on those lists? Or I could also try to lateral in earlier (after say, 6 months at my current bank) and just do a 2nd stint as a 1st year analyst?

Do PE firms look kindly at laterals because I will have more deal experience than a brand new analyst who is just hitting the desk? I'm working at Baird/HL so I am confident I will be on a number of deals, but they probably won't be super high profile or interesting. Or will I be looked down upon because I'll have a less 'prestigious' resume than someone who went to a top bank straight out of school? Am I being insane and should just recruit from my current bank? I really like the people at my bank and would have no problem doing my two years there, but most exits are LMM/MM PE and I want to start my PE career at a larger place.

Just wondering if anyone here has experience with lateraling and then recruiting for MF or other top PE firms and can talk about their experience. Thanks!