Agree with Musken, Ubud has really evolved to tourist central and lost much of its charm. East Bali where Amankila is located offers a much more authentic experience. Lots of beautiful small temples and paddy fields around Amankila (within an hour drive) and Besakih is actually closer to Amankila!
The resort organises traditional dance perfomances regularly and I believe they have recently reintroduced the fire dance! They also have a tour to a traditional Balinese village where women still weave sarongs and craft pottery the old style.

If you really have to do a day tour in Ubud, Amankila is only about an hour away, so a day trip is possible. I think that will save the hassle of packing/unpacking.

As for diving, Kila teams up with an outside diving company (I believe it's called H2O sport) and they have a very detailed diving brochure in room and I'm sure they can send you one via email. I remember a dive ranges from US 150-400/person including transfer to launch site. East Bali has some amazing dive sites but current can be fairly strong.

The only problem is that Amankila is very remote and you certainly won't find restaurants and shops like those in Kuta/Seminyak. Eating out (for dinner at least) is virtually impossible. But as Musken mentioned, the food there is fantastic and they are happy to make whatever you want.
+1 for doing the satay evening and beach dinner for a bit of variety.
Their seafood BBQ was divine

Unless you absolutely have to eat out in evening, stay 4 nights at Amankila and you can do beach, dive, temple/village/paddy fields sightseeing all in one go!

Beach dinner


Classic Kila shot