If you're so astonishingly lucky as to be in receipt of a traffic service ...
If the military airfield beneath you can provide you with a more appropriate traffic service then the people you're already talking to might suggest you change to them.
You'll probably be wearing the squawk of the people you're talking to, so if the military airfield care about you they can tell who you're talking to and get on the landline. I've had this happen to me - I've had a call saying "xxx request that you ..." until a fast grey pointy thing had landed.
You can ask the people you're talking to "should I call xxx and tell them I'm overflying".
Then there's local knowledge ... some military airfields are well known for being perfectly happy to offer a traffic service for miles around, others are well known for impenetrable American accents that you really would prefer not to have to cope with