Welcome rgpbeme!
Welcome to the forum rgpbeme, glad to have you here!...
I know what your talking about and it's happened to mine a few times. It's usually from the battery being to low to draw enough amps to power up the electronics or the lanyard not making good contact, like you said. But it seems this problem she/he is having doesn't seem to be related. Only because it's stated that a new batter (and load tested) was put in and a new starter. It's further stated that it won't roll over with the plugs in but when pulled out, it'll turn over. So that doesn't seem to be electrical.
In the old days, you could retard the timing on your distributer to get a little bit more horsepower but it often made the engine hard to turn over. Though the timing on the Rotax is, for the most part, predetermined, I'm still wondering if there's a timing issue.
If she/he can share a little more info, it may help. When you say the engine won't turn over with the plugs in, do you still get the two beeps from the mpem? Does the engine make any attempt to roll over and just stop? Are there any clicks? When you pull out the plugs, you say it will roll over freely. By description, this is a weird occurance, so I'll be curious to see what you can do to fix it.
Louis