You're pulling a large number of amps from the battery when you crank the motor to start it. That pulls the voltage down to the point where the VHF thinks it's in a loss of power state, which trips the alarm. Essentially, you're power cycling the VHF when you hit the key, which triggers the "power on, everything's OK" beep.
It could be caused by a lowish battery, too-small battery capacity, resistive battery cables/connections, slightly oxidized fuse connections, poor wiring connections, and probably a few other things that I haven't thought about. Easy way to test it is to put a voltmeter on the VHF power input and see what happens when you hit the starter.
I would shy away from an auto-on setup for the VHF running off the motor - what happens when the motor dies and you need to yell for help?