It's been a LONG time since I dealt with an issue like this, but....
I remember on "some" solenoids, there is a internal bypass switch, which was originally provided to bypass the in-line ignition resistor, to pump the voltage up on the Coil, while cranking the engine only.
Could this be an issue of mis-wiring, where when the solenoid engages, it latches the solenoid coil closed, and the only way to release it, is take the Batt terminal off.
That's a long shot !
Have a good Day !
S.W.
PS: I just thought of something else...
Try reversing/switching the Battery and Starter Main Cables on the Solenoid (large posts).