> My employers were somewhat negative about the idea and suggested they might need to bring on a freelancer to cover my position for a few of the days I'd be out

Really..? They can just pull in a freelancer like that? I have a hard time believing it... Sounds like an empty (and bizarre) threat to me. Either way, just say "great pull in the freelance for 3 days so I can take my vacation".

> Is there any way to reason with them?

I think the only thing you can do here is take your vacation. Stand your ground and do it. It might feel awkward but if you don't they will continue to push back on your vacation time.