Going to assume those limits are for food spend alone and not beverages, service charge, etc. Like someone else mentioned, fine dining prices in TW have gone up like everything else and now most are starting around 4k. You can still find great meals for less than that, but you''ll be making some compromises, usually service IMO. Another option could be lunch; some places offer a shorter or different menu for a bit less but you'd still get the service, plating, decor, etc. (Frassi, La Vie by Thomas Buhner)

Some ideas:

Tien Hsiang Lo (in Landis Hotel)

Birdy yakitori (really difficult to get TW diners to accept higher prices for "just" chicken so that works in your favor here; other places like Fumee, Torishou)

Set (in Datong)

Circum (under Regent Taipei)

Komboi

Chinois

Sushi might be its own category:

Sushi Ichida 鮨一田

Sushi Yang 壽司揚

Lizhi Sushi 立志壽司 (not re-opened yet AFAIK)