Over the years I have found two part epoxy the best solution for strong wood repairs. Epoxy your replacement piece on adding wood screws like you stated above and then once dry fill with additional epoxy and sand as smooth as possible. Once finished you can use a little bondo or filler to finish things off before painting.

Why worry about Taxi logo at all? Just paint to bottom of coin door yellow then get a good picture of the front of a taxi and have sign shop make you a decal. If you prefer painting it just make a stencil from a good picture and you will be good to go.