I've never stayed at Amandari, but my mother was there about a year ago and had a fantastic stay. Not much more I can add but she said it was very peaceful, quiet, and relaxing. Not sure if that is the vibe you are going for.
When we went to Ubud in 2014 we stayed at Chedi Club and I'm not sure if I just had love goggles on (it was my fiancee's and I first real int'l vacation as a couple), but I would rank it as one of, if not the, best hotel stays of my life. The service, F&B, room, inclusions (laundry!), the only time butler service has ever actually "worked" for us, and the grounds all made it quite special. Not to mention it is priced at a steal IMO.
We are planning our return to Indonesia next year and if we were doing Ubud we would most likely do Chedi Club, even though we are doing Amanjiwo and Amankila as well. Part of us wanted to do Amandari just to make it seamless between Aman's, but like others have said, we don't really want to go back to Ubud as it felt quite touristy. Instead we will likely do the Legian in Seminayk, as it has a bit of a lively atmosphere which will hopefully contrast nicely with the "quietness" of Jiwo/Kila.
From what I've read here, Amankila does have a bunch of cultural experiences on offer, and they may be more authentic than that of Ubud given the tourist surge. Though I do find it hard to believe that even with the touristy nature of Ubud there is more to do in East Bali.