As a British citizen with a PRC wife with British Permanent Residence and a PRC passport, would she be able to enter and live in Taiwan if I was to get a job there?
This may depend on multiple factors. Suggest you email the Taipei Representative Office in London, explain your situation and ask for guidance.
Include all the details: what job, company, contract duration, salary, what visa/ARC you expect to have, where you are currently resident, how long you have been resident there, how long married, marriage registered in which country, wife's household registration details in China, . . .
Be concise and courteous.