OK,
Your jab alternator is indeed an AC device. However I point out that the Jab regulator is not what one would call "automotive' type. Maybe motorcycle type. It has no form of current regulation, and due to the lithium battery having a amazingly low internal resistance, it is quite possible that the Jab alternator/regulator combination is overloaded during climb out. After all, the battery has just discharged significantly during cranking, the motor is running at high rpm, so the alternator is putting out max power,and the regulator won't be cutting out until battery voltage reaches its fully charged voltage. Until then, the battery will be fully loading (maybe overloading) the charging system.
In the absence of a ammeter, you can't prove or disprove my theory.
This is the 'problem' when using a lithium battery in a system that was designed for a different kind of battery.
It needs a regulator that limits the voltage and the current. I am not aware of any. BTW, this applies equally to Rotax, as they have a crude Ducati system similar to Jab.