VP shortlist at current shop vs. signed BB offer — thoughts?

Looking for outside perspective.

Background: about 3 years into equity research at a middle-market shop. Signed an offer a few weeks back for a Senior Associate role at a bulge bracket, joining a ranked/well-regarded team in the respective sector. Offer is signed, but my start date has slipped due to some paperwork issues. Shortly after signing, I found out I'm on the shortlist for VP at my current shop, comp at VP there would land roughly in line with the BB offer, so this isn't really a near-term money question, more a platform vs. title tradeoff.

A few things I'm trying to think through:

1. Given comp lands similarly either way, is there a real case for staying to chase VP at the current shop instead of the BB move? Curious how others weigh title now vs. long-term platform.
2. Is it reasonable to raise the VP shortlist with the new firm at all, e.g. asking if there's a path to be considered for their own cycle sooner given where I already am? Or does that read as presumptuous before I've even started?
3. Any thoughts on handling the optics of a delayed start date plus a live promotion elsewhere, without it looking like I'm hedging or shopping the offer?

Appreciate any perspective, especially from people who've navigated something similar.