"With the Doctor's help, these bases would exist beyond space and time, so all spaceship can reach them and attack them."

Fuel and attack don't matter for dummy bases :)

The two alternatives are:

1. You do nothing, as in you build zero dummy bases. In this case, it's clear why the answer is maximum matching * k

2. You build enough dummy bases to guarantee that all ships that were attacking real bases will now attack dummy ones. In this case, it can be shown that you need to build s dummy bases. This is because unpaired ships will be paired initially with dummy bases, then the ones in the matching.

Doing anything in between is guaranteed to be less optimal.