Battery replacement

The reason your electronics are dropping is because the voltage applied drops below the required amount as when the batteries are under heavy load of starting an outboard engine is applied using a marginal battery. Yes, you can do all the testing recommended above but the simple fact your batteries are 5 years old is enough to recommend battery replacement, this should fix your electronic drop out issue.
 
It would help to know what the voltage at the electronics are with the a battery turned on and the then the electronics turned on. The second will be lower but shouldn't be a lot lower. Then turn on the ignition without cranking the engine and se what you read at those electronics again and if possible what the voltage is at the battery. Again there shouldn't be a big drop but I suspect you might see one.