I just have tried it, and ATC given me clearance to start up, when I requested start up with engines running.
To make it working I did the following. I placed friendly ground unit somewhere on the airbase, to make the base friendly. Then I started the mission (take off from runway, Batumi), have set VHF AM to ATC freq, took off, landed, taxied to the parking area, pressed LAlt + Num+ to call VHF AM menu, then proceeded to request start up.
What key you were using for activating radio menu? Backslash of one of Throttle mic switch keys?