You have to "force fire" on the spammer station and destroy it. This is the same with the USS destroy pirate bases as well. To force fire on a neutral target you press and hold control while having your ships selected and then right click on what you want to shoot.
Targeting can be a bit wonky in SW 2 so if does not work on the main screen try force firing from the tactical map, it's easier sometimes.
Hope it helps.
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