Well there are a couple of things, going through all the battery connections and checking them also at the solenoid is good, and I do believe your battery is OK. If you notice there are a few black connectors where a bunch of wires go into them, those are referred to as "joint" connectors, what they actually do is form a connection for common wires, what can happen is moisture can get inside and then foul up the connections inside them, you can pop the cap off and inspect for corrosion on the terminals, clean then and then reapply dielectric grease back in the connector and see if that helps.
Next back by the battery is a 16 pin cconnector that connects the "main" harness to the fuel injection harness, pull that apart and again look at all the connections close to make sure nothing is going on with them, get a can of contact cleaner and spray the connections, you have an intermittent problem, its likely not going to jump out and bite you, this could be a process of elimination to say the least, I believe on my past experiences that you will likely find an issue in one of the harness. The down side of all of this is you do use the ski in salt water and as good as you think you are cleaning everything, salt water works its way into these connectors and even the wiring harness as good as we think they are sealed, they are not in the adverse conditions.
Let me know of this help, if not I will be glad to answer anything else.