Does the Galactic Government need a military: NO

Why?

Because it has the Galactic Time Guard for that which can disappear the whole galaxy from time if necessary while changes are made if necessary.

The most powerful being in executing this is the Galactic Sentience's consort Purple Delta 7 who is a 5 to 7 million year old sentient android robotic form (basically at this point she can literally be anything from an atom to the size of a galaxy with her present technology allowed by the galactic sentience in place.

And she keeps thousands to millions of copies of herself like beacons or GPS satellites sending back information to her from thousands of planets, time lines time zones etc.