Doing a second pre-mba Associate stint at a larger PE firm

I work at a lower MM PE firm and am wondering if it is worth exploring this option once I finish my current program (2-3 years). Let's assume for the sake of the discussion that my long term goal is to end up in a partner-track position at as large a PE firm as possible.

My reasoning for considering this option:
1. Increase long-term firm size: Once you get to the post-MBA level, PE lateraling generally moves from larger firms to smaller firms, so going into an MBA program from a larger firm would in general increase the size of the firms I could potentially end up at down the line.
2. Brand: potentially more recognizable firm name would likely help with post-MBA recruiting (and depending on the firm, MBA placement)
3. Deal experience: PE recruiting at MBA programs is competitive, and having a arsenal of deals to talk about can be a point of differentiation. Doing a second stint would not just give me more deals to talk about, but potentially more experience in different parts of the process, since firms differ in the exact responsibilities in the Associate role
4. Increased potential to be a top performer at second firm: all things being equal, taking a "redshirt" at a smaller firm means I'd likely be coming in with more experience than other pre-mba Associates- have reps under my belt, be more polished professionally, etc.
5. It's only another ~2 years: in the grand scheme of a 40+ year career, who cares. If it puts me on a better trajectory, I'd think it could be worth it.

My questions are the following:
1. How would recruiting work to move from one pre-MBA position to another? Would HH be receptive? What about PE firms themselves? Could I basically be treated similarly to a PE Analyst?
2. Would a move like this look poorly to MBA adcoms? What about PE firms down the line? I know it could look like a lack of career progression..
3. Coming from a lower MM firm, what size firms would be possible? Core MM? Upper MM? MF?
4. How would senior folks at my current firm likely react? It's standard for people to get an MBA after their program, with a small % going to other roles or getting promoted straight to a post-MBA level, so don't know if they'd be supportive.

Interested in any perspective. I didn't find any threads on WSO on this exact topic, but if I missed any please point me towards them. Thanks everyone!