- Which mountains should we aim for?
There are many, many to choose from, but keep in mind that you are going in the heart of the wet season. Many may be inaccessible.
- How easy is it to get from Bali to a worthwhile mountain to bag?
From our trip planning, it seems pretty easy. In fact Bali itself has nice peaks, though places like Batur are touristy.
Java is a short way away and has many mountains. Lombok has many peaks as well.
We're going to try and climb Gunung Lawu near Solo (Java) since it's supposed to be OK in the wet season. The peaks in the Ijen Plateau area are in far east Java right next to Bali and are supposed to be reasonable in the wet season as well. The most spectacular peak in the area is supposed to be Sumeru, but it is also active and may be problematic in the wet season.
On Bali itself, Gunung Agung (3142 meters) is the highest point and it sounds very interesting, but is supposed to be a trial in the wet season. Gunung Batur is supposed to be reasonable in the wet season, but it sounds quite touristy so you may not be interested in it.
Lombok has mountains up to 3726 meters which is quite respectable for an island.
If you want a real adventure you can fly to New Guinea where there are plenty of spectacular and mostly unexplored mountains, though it would probably take more time and gear than it sounds like you have available.