MC Starting Problem-Please help!!!

I have been having ongoing problems starting my MC so anyone please help a bro out!!

When I put my boat(99' MC Maristar) in off the trailer it starts up fine, we cruise down the lake to the good water and kill the engine to fill ballasts etc.

I try to start it back up and it wont attempt to crank. The volts are good (12+), it seems a lot like when the throttle is left in drive and you try to start it--but it's in neutral!! Then, after a while (say 10-20 minutes) it will start back up ok.

I replaced the battery and alternator but the exact same thing happened the next time I took it out. Then I left the boat sitting for a few weeks and when I went back today the new battery was dead.

From posts on here I have narrowed the problem down to either 1)solenoid in starter (I think it is probably this one) 2)MC ECU box or 3)loose wire.

The only other pertinent details that come to mind are it has made a clicking noise (although not always) when I try to start it, and I recently installed a new amp (which I cannot think of how this could be it but nonetheless . . .)

Any suggestions to help me tackle this challenge are very appreciated!!!! Thanks.