I'm not sure when you are travelling but there are two hotels I recently stayed at with (younger) kids: Hilton Bali Resort and Courtyard Bali Seminyak Resort

If you don't want to read these reports, the summary I think is that the Hilton felt a bit dated (there was some renovation work going on) but was right on the beach while the Courtyard Seminyak was clearly newer but wasn't right by the beach (both of these have kids clubs). The Hilton has a better pool for older kids.
The spa at the Hilton is open air, so if you think you want to have air con, try the Courtyard Seminyak - those these are just hotel rooms that have been converted.

There's also a Courtyard in Nusa Dua which I also stayed previously (no report) which I personally really enjoyed but again not by the beach - though they have a free shuttle that will take you there.

With respect to be vegan / vegetarian, there should be no issues with this at all. I've found all the major hotels to be able to cope with this without skipping a beat.

Finally, you should take a look at pricing and see if there are any special rates available for the dates you are looking at. For example, some hotels offer, pay 3 stay 4 nights or something like that.
Oh and really finally, just take a look at the Mulia Resort - this is definitely on the luxury side and you might find the pricing here is a higher than other places but for some dates might be reasonably competitive.