The charger will charge both batteries to full capacity as long as it is installed correctly. (If it wasn't installed correctly you would know due to dead batteries or blown fuses by now.)
Your concern about the weak battery is not exactly correct. When you have two batteries connected in series, the total current output will be limited by the weaker battery - but it will not harm, damage, or otherwise change the stronger battery.
Ex: I have one 200A battery in series with a 100A battery. The series combination will be limited to 100A current output, because both batteries must be passing the same amount of current. In all likelihood, the 200A battery is larger, so it will retain charge longer.
This just means it does you no good to have one big battery in series with a small battery. But they will both fully charge and one won't damage the other.