When you want to take a test or enter a competition, the papers are supposed to be signed by an Officer, so you'd have to manage that long-distance. I don't know if the new club chair can give you a blanket "anything she wants to do is okay" letter, but it's possible. Watch deadlines: getting a signature at the last minute might be difficult. Our club has gone paper-less, so signatures are only an issue if you want to test/compete elsewhere. In a pinch, an email/scan from the officer to the other club's chair could suffice, with the original to follow. You seem to be on top of things, so I'm sure this won't be an issue.
You wouldn't have voting privileges at your current club for officers or proposals, but you might/might not at the new one.
I don't know if that matters to you, though, unless you're active in one or the other and want to have a say.
Skater support might be an issue, if your club offers it and you're eligible. My kids and I received a check for competing at Synchro Sectionals last year, and for testing.
OT: When the USFSA raised the Individual Membership rate, I wondered if anyone chose to join a club in name only, to save money. I'm sure that was the intent behind the 40% rate jump. Our club rep asked about that and was assured it was "only" 16%, which was untrue. He didn't make an issue out of it at Governing Council for that reason. Maybe someone was looking at the wrong line on the chart.)
I could see an enterprising club (that needs members) offering a bare-minimum rate to long-distance members. It's a win-win: save the skater some money and boost the membership count for the club.