[Uhehner]

Rule number ONE : Never leave the boat with "anything" ON unless you are connected to shore power and your battery charger works.

Never anchor out with anything ON except the anchor light.

Have a GOOD mechanic check that each engine will supply about 13.8 volts to its battery with the battery charger OFF.
Next stop the engines and check that the battery charger supplies 13,8 volts to ALL batteries with the battery switches OFF.
Avoid putting port and starboard  batteries in parallel.  If one is bad they will all be dead.

Why did you need an external battery charger? Is the one on board also bad?

Your report says to me that the starboard alternator is not putting out and its engine only runs because the battery switch is paralleling the P&S.

If you install AGM batteries, make sure your battery charger can handle them.

With a new boat and 3 year old batteries it's time for a full replacement. 

As a starting point contact Tiara for any boat parts you need and Cummins for an alternator.

Welcome to the world of boating.

Capt. George