Experimented a bunch more, the "Instant" tag doesn't work in this instance. What I will probably end up doing is modding the Starbase modules with something like:
<Stats>
<EffectType>StrategicRepair</EffectType>
<Target>
<TargetType>Starbase</TargetType>
</Target>
<BonusType>Flat</BonusType>
<Value>25</Value>
</Stats>
This won't instantly heal them, but it will do so over a few turns. Stylistically it reflects that you made this super-spiffy addition, but it'll take a few turns before the Death Star can achieve full operation.