Re: Do these engines have charging systems?
Most (nearly all) outboard motors just don't have the capability to replenish an Ah draw that big. The isolator is a great idea if you have a 60+A alternator such as an I/O or inboard, but it is useless in your current setup. Ditch the isolator and run the stereo off a dedicated battery that gets charged either by an onboard 120v marine charger or on a charger in your garage. I have my fishing boat set up to run everything including sonar, interior lighting, livewell pump, and a stereo with amplifier off the trolling motor bank. The only thing the outboard battery runs is engine loads, nav lights, and primary bilge pump. Each night the boat gets plugged in and recharged via the onboard charger.
Edit: Do not use a plain jane automotive charger to recharge your batteries. Use a 3 stage charger that has automatic bulk/absorption/float. Using the typical 10A/2A auto charger is pretty much guaranteed to boil off the electrolyte in your batt and if you're using a gel or AGM batt it will not be outputting the correct voltage to prevent premature failure of the battery.