12.7 is not enough. Maybe the converter needs to see a battery before it goes full charge. Try disconnecting shore power and then hook up the battery. Take a voltage reading across the battery terminals and write it down. Then plug in shore power and take another voltage reading. You should be seeing at least 13.2 volts if the batteries are fully charged.
Check the voltage of the starting battery or the starting battery of your car to make sure your meter is working.
Give us the readings and we can tell you if it looks like anything is wrong.