Lateral from BB to EB after signing PE offer

Will be starting in a top BB group that places well into PE which is my goal, but I've always been a bigger fan of EB model (i.e. owning the entire process start to finish, leaner deal teams, higher pay, generally better cultures). Not sure how early on-cycle recruiting will kick off this year (has it even begun for 2020 class?), but assuming it reverts to a pre-COVID schedule, would it be possible to lateral banks after signing a PE offer?

For example, my timeline would look something like this: 

August 2021 - hit the desk at BB

October 2021 - on cycle kicks off and I sign a buy-side offer

January/February 2022 - lateral to EB

mid 2023 - start at PE firm

I imagine the PE firm wouldn't really have a problem with it, assuming I disclose everything to them, but how would the lateral process work? Would I even be able to move banks given I've already signed an offer elsewhere or would I have to disclose that? 

Also, if PE recruiting starts later (like January 2022) would switching firms be viewed negatively? My group does a lot of M&A volume, so I should have at least 1-2 closed deals by that point, but at the BB.

And is it even worth it to think about switching firms? My view is that this is 2 years of my life so I should optimize for the best overall experience, but do I just have a grass-is-greener mentality about this? Can anyone who has made the jump opine? Thanks