A winter has passed since I updated this thread.
Last September, I had a few new occurrences of the problem after replacing the DESS post late July (ordered directly from a BRP dealer). The next day, I took my spare key (new) and I rode a bit on the lake without being able to reproduce the issue. But then I had to take the Sea-Doo out of the water for winterization.
So today I finally opened the handle bar cover in the garage to check the wiring connections that I made last year when replacing this DESS post as I wanted to make sure that my connections had not taken water. Last year, I had to cut and splice the wires to replace the post. I then inserted wiring shrinks before reconnecting wires (no welding). Then I used the shrinks and then I taped the whole thing with electrical tape.
As soon as I opened the cover today I immediately saw that no water had been there. Everything looked A1. I still removed the tape and saw absolutely no signs of humidity and oxidation. I did not break the shrinks because it had been difficult enough to put them there in the first place last year (not much room to work and lots of plastic surrounding the wires so I had to be really careful when shrinking the shrinks).
So using both keys alternatively (the old and the new), I connected them on the post (battery connected) and moved/twisted/pulled each wire one by one to see if the beeping would stop but no, I wasn't able to reproduce the issue of last September. The new DESS terminal still also looks brand new. Checked all the wiring leading to the MPEM and didn't find anything corroded, loose or anything else.
Anyone has more suggestions for me? Could it be the MPEM showing signs of fatigue? Last year, it was definitely the DESS post because I had the ski working correctly for two months before the issue started to come back again. Last year in July, it had come gradually and up to the point where the ski became unusable. The I replaced the post.
So I am not sure what else I could test today.