A possible POD could be an earlier discovery of gold in Jiufen (maybe in the 1840s), with British commercial entities going there on invitation by local authorities or even just local merchants. Then over time that could develop into some concrete British interests in the region, just in time for the Second Opium war to break out.
There wasn't a gigantic number of Chinese on Taiwan at this point so it wouldn't have been ungovernable (and if the occupation of Zhoushan during the 1st Opium War is any indication, malaria killed more Brits than Chinese resistance).
Well technically for a while Hong Kong was the colony of Victoria, so the British will want a Western name for 'Taiwan'. Formosa works, though, though you might have to give a different name for the then-primary city, Keelung (maybe Palmerston or something).