Where is this? In my experience in your scenario the civilian ATC would dump you and tell you to freecall "whoeveritismatz" unless they have an agreement with.
Anyway assuming they don't, if in receipt of a traffic service I'd just carry on with my routing, as you'd be squwarking, and no doubt the mil could see you anyway.
However if NOT in receipt of a traffic service, I'd call the matz radar controller and ask for a traffic service. When he asks your routing just tell him / her you'd be routing via their overhead but remaining clear of the MATZ.