Jules,
I definitely recommend Caffe Florian! When I was there a little over a year ago, they didn't take reservations and getting a table wasn't a problem. I had a couple questions about timing as well since it was March, and they were very responsive via email so you might want to reach out for specific timing info for your dates.
There are fees for sitting with the music playing, in addition to the menu costs, so I'd recommend looking over the menu ahead of time if you're at all concerned about cost. There were many people who sat down, looked at the menu and unfortunately left. We went in fully knowing it would be expensive and budgeted 100 Euro for the evening. After two soft drinks, some macarons, and scones with cream and jam, our bill came to maybe 70ish Euro if I remember correctly.
Sounds like alot, but we spent a good two hours being serenaded there as the sun set over the square and it was worth every last cent to us. There may have even been happy tears it was so special. It is one of my most favorite memories of our whole trip!
I hope you have an equally magical time!
PQ