Why would the Qing invade a trading post that supplied them with all that mexican silver?
Spanish coins were widely used in China. If they alienate their relationship with Spain by invading Spanish Taiwan they would lose that silver.
Why would Spain need Taiwan as a trading post to China when they already have Manila? Besides, when Spain and Portugal were in a personal union there already was access to the much convenient Macau. And what natural resources does Taiwan provide which the much larger Philippines (or for that matter, the much closer Central America) cannot?
Let me reiterate my main point: any European colony on Taiwan would almost certainly attract Chinese settlement much earlier and much more numerous than OTL. Given Spain's hostility towards non-Catholic religious practices (or anything else which impedes the life of the Chinese settlers), a Chinese rebellion will happen sooner rather than later. This does not mean the uprising occurs *because* of the Inquisition, but the Inquisition plays a role which would not exist if the colonial power was tolerant of religious diversity. This in turn could lead to a voluntary annexation by the Qing Empire. Even if Taiwan remains independent, life won't be easy for any (remaining) Catholics on the island.
And finally even if Spain is annoyed by an actual Chinese invasion of Taiwan (which seems very out of character for a civilization which created The Art of War), would they end the extremely lucrative trade route over it?