Is there a way to look at an open invoice and know if it's already been sent to a customer without looking at the Communication tab?

Without a strong process in place or custom fields to track this (and a process) you are pretty much in the dark...sorry for the bad news but I've been down this rabbit hole

What I ended up doing was adding search criteria to exclude any invoices with messages on or before today. Seems to work so far.

Where I used to work would have blown that up...lots of chatter on transactions

replying to old message, but the search criteria could filter out communications to @[companyemaildomain]