Many thanks for reading! You're absolutely right that Bali's beaches aren't objectively the most beautiful out there. In my opinion, the beaches in the southernmost portion of the island are the most beautiful, featuring pristine white sand. As such, considering the beach alone, I'd recommend staying in the gated Nusa Dua area, given the area's pristine stretch of beach and the fact that it is swimmable. Although be cautioned that it is a luxury hotel compound, so it's not a very cultural experience (if that's what you're after). I would, however, particularly recommend Amankila, which despite being further away, is an absolutely lovely hotel. Its beach isn't the most beautiful but it is private and it the hotel is an all-around great experience.

As for Ubud, I think the three places you're looking at are all very great choices, so you won't go wrong between any of them. Between the three, I've personally stayed at the FS Sayan, and have visited Mandapa and Viceroy on a couple occasions. I personally really liked the FS and I found its architecture and service to be outstanding, and their river pool to be beautiful. However, the hotel does not offer much as much of a view (whereas Viceroy and Mandapa features valley views) and their food is forgettable. Viceroy is run by an Aussie family and has a great pool, but its rooms are rather aged. Mandapa is the newest of the three, and I personally like the hotel's design. One thing to note is that both Mandapa and FS are connected to the Ayung River where many rafting activities go through, so occasionally you might hear screaming rafters when staying at their river villas.

Safe travels!