Mpem ?
I was going to stay out of this one since it appears, KustomKarl has given you all the test procedures that you could possibly use to find the answer.
I think your mpem is bad. Not bad in the sense that it won't allow the engine a start, but in it's circuits that direct its functions to work properly with the electrical system. If you can jump the solenoid to the battery and the ski runs, then the mpem signal circuit to the solenoid from the start/stop (double acting switch) and the anti theft circuit (DESS) is fused. The other side of the start/stop switch is routed to ground your ignition system to shut down the engine.
Unless someone has gone in and bypassed the electrical switches in some manner, looking to go around the anti-theft security, that yellow/ red connector wire should not have any voltage. The only time that it should have voltage is when you put your lanyard on, you get the two beeps (to signal proper mpem function for that ski) and then, when you push the start button, the signal from there, runs through the mpem, then out to the yellow/red wire for power to start and run the ski. When the DESS cap is removed, that power should go off.
I'm not you, but if I had to identical skis, one works well, the other does not, I'd take the DESS post and mpem from the two skis, disconnect batteries first. Then put the post and mpem from the one in question on the ski that you know to be good. Then connect the battery. Test the yellow/red wire. If it's hot on that ski too, then you know it's the mpem. You can now, put the mpem and DESS post (known to be working correctly) in that ski you had the problem with, and if it's not hot at the solenoid, works correctly, then you'll back up your findings on the mpem in question.