How to get more interaction with PE clients?

Currently staffed on a transaction in which we've been mandated by my "dream PE fund." The deal team is quite large, with several people at the senior / VP / As level, and therefore there are several lines of communication for the client before reaching me (1st year coverage analyst).

I would ideally like to create a rapport with key team members on the client side so that I can be at the top of mind when PE recruiting season comes around, but finding it a bit difficult with such a thick deal team / no in-person meetings due to COVID. The only good news here is that the coverage team is holding the model rather than M&A, so I'll be managing that along with an associate over the life of the deal.

Have any of you guys been in a similar situation / have any advice about gaining more client interaction to hopefully exit to a PE client? Is it common for IB analysts to exit to PE clients that they work with?