someone's been playing eu4
Well, I did some research on a similar topic earlier, and...
Ryukyu is dotted with fortresses called gusuku
https://en.wikipedia.org/wiki/Gusuku
which are one of the legacies of feudal war among feudal lords called anji or aji
https://en.wikipedia.org/wiki/Aji_(Ryūkyū)
These anji were brought into line and subordinated to royal authority by King Sho Shin that was organized into units and subunits.
http://wiki.samurai-archives.com/index.php?title=Military_of_the_Ryukyu_Kingdom
The article that that SamuraiWiki article cites states in the mid-1500s (Sho Nei's reign) that there were 3000 soldiers and anywhere between 50 and 100 ships.
"In terms of the size of Ryukyuan armies, documents connected with Okinawan invasions of other Ryukyu Islands, mobilizations to defend against pirates, and the mobilization to defend against the Satsuma invasion of 1609 indicate a range of between 1000 and 3000 soldiers, with naval flotillas ranging in size from 46 to 100 ships."
Population stats on Taiwan are few and far between but this Wikipedia article estimates maybe 100,000 people before 1600
https://en.wikipedia.org/wiki/Demographics_of_Taiwan#Historical
So, 3000 people holding down 100,000 people, across the plains and mountains of Taiwan, is a bit much with some kind of Ryukyuan population wank.
You could get extra population, however, by having the RYukyuans adopt a piecemeal effort at colonization:
1. Building gusukus along the western plains to house farming settlements (essentially walled cities)
2. Try to assimilate natives into the Ryukyuan peerage system, either with trade or offers of protection.
3. Use the extra population gained from Steps 1 and 2 to conquer even more.
Of course, this is dependent on the Ryukyuans not getting swatted immediately upon arriving or bankrupting themselves in the process of colonization later on...
The future of a Ryukyuan Empire might not be in Taiwan, but in Polynesia. I think that Ryukyuans ships made it to Thailand at some point, I'm pretty sure that they can make it to Palau and beyond.
The real challenge, though, is getting Japan to stay out, because the more Ryukyu grows the more appealing it starts to look to Kyoto... or to any of the outlying daimyo capitals, for that matter.