First, almost any live aboard will be a step above Blackbeards, if not 2 or 3. I've been on some crap boats that would seem very roomy compared to Blackbeards.
Ease of travel: I'd go with either 1) Fiji on the Naia or 2) High end boat in Thailand.
There are no other flights after you arrive in Fiji in order to board the Naia. You would likely fly to Bangkok, then to Phuket, but it's a pretty easy transfer. Overnighting in Bangkok isn't expensive and spending a bit of time in Bangkok is worthwhile, political unrest not withstanding.
Sorry, I'm just not that in to Palau. You can get there in one straight shot, but check out your routing. Yours would be shorter than mine, but I went Dallas-Houston-Hawaii-Guam-Koror and had plenty long airport layovers in each. I arrived each way near mental death, although a shorter trip miles wise.
I found Thailand least drifty, but each of these destinations has a multitude of sites that have either strong or weak currents. Lunar cycles, tides and site geography all play in, so levels of current can vary. I've dived Blue Corner in Palau with absolutely zero current, which meant there was nothing to see.
BTW, I think there are like 8,000 liveaboards in Thailand. Some are figuratively junk. Some are literally a Junk.
So, it is possible to find something a step down from Blackbeards. Seek professional help in your selection.