I have a 12V starter motor that needs extra help at times.
To me it seems like the battery can't deliver enough current to the starter motor. I would not try to raise the voltage since the starter motor is rated at 12V. When the battery is connected to the charger the charger helps the battery to supply enough current. To be sure I would look at the starter motor specification. Look for the wattage or current specification. Also look at how much current the battery is specified for. The battery must be able to deliver as much or more current than what the starter motor needs.

Two solutions to the problem if my suspicion is true:


Change battery to a model that can supply more current.
If you have the space, add another battery of the same kind as you already have in parallel. That way you will be able to supply the starter motor with more current and keep the voltage at around 12V to 14V.