I am currently on a 60-day non-extendable visa from Britain. It is due quite soon, and I would like to apply for another visitor visa to come back to Taiwan. If I get an invitation letter from a local living in Taiwan, is it likely that the visa office in Hong Kong (that is where I intend to go to get my new visa) would reject my application? I already received an invitation letter from this friend and the London office provided me with my current visa. I have an onward ticket, enough funds etc.
Are you sure it is non-extendable? 60-day visitor visas are extendable unless they are explicitly stamped as non-extendable. Unless you are from a southeast Asian country, visitor visas issued by a representative office are usually the extenable kind. If it is not stamped as non-extendable then you should be able to go to the National Immigration Agency to get a 60 day extension. A 60-day visa can be extended twice.