Or you could just build an airstrip on those territories and station some land-based planes there.
I also don't think it's unreasonable to consider it. We also considered with the Eurofighter but then decided against it (which is one of the reasons France left the project and build the Rafale instead). It's just not a capability that most nations need so it makes sense that they don't want to invest in it.
I feel this is a bit backwards. Britain can get away with relatively few defense on their oversee territories because they have aircraft carrier power projection. They could still have won the Falkland wars without carriers if they put a proper military force on the island to begin with.
Also the current biggest threat and main reason we need this new jet is the Russian threat to mainland Europe. And I don't see the 5 European aircraft carriers that currently exist playing a major role here (and 3 of them already have F-35s anyway).
So let's get the plane we actually need first. I'm sure France's oversee territories will be fine with just the Rafales protecting them for now.