A low Battery (anything below 12volts) should instantly go to 13.5- 14.2 volts dc the second you plug in the shore power cord. just watch the voltage difference at the batteries... unplugged verses plugged in rv cord.
I doubt that your convertor would be only a 2 amp charger. I know it better to charge batteries at a slower rate but that is rediculoius. (sp.)
My charger is a rated 20 amp and rarely see both batteries able to be force fed more than 16 amps at a time.
Look at small print on your convertor to find actual rating. These days I need two inch thick reading glasses and a flashlite in daytime.