The cheapest Aman-suites in Bali are no more expensive than the rooms at the Bali FS resorts if you look at the rates they list at their internet sites (typically USD 750++ per night - do not know about differences if you book though a TA). You do get a private pool at the FS resorts for this price per night, but I would rather take an Aman for the total experience (and Aman is more willing to upgrade people in my experience). It is so special and magic to be in an Amanresort, just perfect for a honeymoon. Amans offers a number of packages as well which you really should take advantage of if you stay there - the Aman activities/excursions are really something special.

But of course: Budget is important, but since you say that FS would be OK, I just had to say this.