What is the best way to hold an aircraft on a base permanently, without it coming off the base? Now I know that means you won’t see the underside very well but can you still put models in a show when they are permanently cemented to a base?
One way to do it is to use your pin vise or motor-tool to drill a very small hole through the bottom of the wheels. Insert a short length of thin brass rod (or sewing pin with the head clipped off, etc.) with about a half inch remaining. If the hole is the right size you may not even need glue, otherwise a little CA will do it. Drill corresponding holes in the base and put the remaining end of the rod into those holes with some CA.
I usually use an unpainted pine base (many sizes and shapes are available cheap at Hobby Lobby.) I finish it with wood stain and polyurethane varnish. Then I get a small plastic or brass rod (threaded at one end if possible), drill a hole for it in the bottom of the model between the wings where it is less visible, drill a hole in the wood base for the threaded end, counter sink the hole for the nut on the bottom of the base, and put a little super glue in the hole in the model. Sometimes I will then mount the base inside a plastic display case, using screws from the bottom of the plastic base into the wood base. Haven’t had one fall apart yet.
But as for putting them in shows, I have no experience with that.
Here’s one showing a wooden base mounted on top of a plastic base for an Imex display case-

I don’t think you’d have any problems at a show. The mosr handling I’ve seen judges do is just, very carefully lifting the model and turning it for another angle. Have never seen one flipped over, they all have flashlights and dental mirrors to inspect the bottom.
I always try to put my models on a base(for shows), just so the judges can turn it by the base and not have to touch the model.
Thanks for the advice.