The battery may have a cell shorting, you can take that to an auto part store and
get a free load test to verify.
The most common reason the battery light stays on is that the rectifier/regulator assembly in the front grey box needs replacing.
And as that said, you should first test the charging of the good battery.
Step #1 is without the motor running, measure voltage in your battery. A simple voltmeter is available in every $10 multi-meter that is also a usefull tool in many ways solving electrical problems on your Sea Doo.
#2. knowing the static voltage in the battery now from your testing, start the engine and re-read the voltage at the battery at 5500 rpm or somewhere near that rpm if you do not have a tachometer. A Sea Doo engine like yours idles at 3000 rpm out of the water, so cracking the throttle a little bit will add 2500 more rpm in a heartbeat. Just keep it off the rev limiter.
If the voltage reading is higher than it was in a static test, but less than 15Vdc, your battery IS charging and that light should not stay on. FYI, all gauges and lights come on after installing the lanyard to show you they work, but idiot lights should go off as soon as you start the engine, or if you leave the lanyard on without starting the engine, 33 seconds later everything electrical will shut down automatically. To re-activate the electrical system, unplug and reconnect the lanyard. Your MPEM is bad, so in your case, the magnito plug.
If your charging system is not functioning, the most common problem is it needs a new rectifier/regulator assembly, about $50-$60 new. Simple to install as all 5 wires that you disconnect are color coded, 2 are yellow, 1 is yellow with a black tracer line, one is red, those 4 have pull apart bullett connectors. The last wire is black and is always the ground and has a ring terminal attached at the end of the wire. One 10mm screw holds it secured to the grey box, and another 10mm nut secures the ground to the central grounding post where you will find other black wires connected for grounding other items in the box.
Sometimes there can be other reasons that a charging system shows an idiot light. These are from a battery with a broken cell inside of it that intermittinly becomes detached from shaking ( not very common sitting on the trailer ). If the stator charging coils are damaged or shorted out, it will not produce power to send to the rectifier/regulator where it is converted from AC to DC Voltage, then regulated, then sent to charge the battery as the battery needs it. When the stator shorts out or is damaged, there will be a blown fuse on the face of the MPEM.
But one thing is constant when the charging system is working, there will be more voltage going into the battery than it had when the static test was taken, it may not be much more, but it will be more. How much depends upon how low the battery voltage gets.
Any rectifier/regulator pushing out 15 Vdc or more to the battery is toast, as well as most that cannot put out anything, The exception to that is the problem lies in the magneto housing or the wiring.
But by far, about 95% of the time the rectifier is the culprit. Bills86e