Larry, since your are permanently plugged in to 120V, your converter should supply plenty of 12V to keep it running. If it is the 40qt., the web site says it draws 5 Amps. Even the smallest converter should handle that easily. The only thing I would caution about would be if you have a WFCO converter, get a DC volt meter, permanently mount it somewhere easy to see, and wire it to the output of the converter. It should read somewhere around 13.2 - 13.6 volts all the time. If it doesn't, start checking things because the converter may have failed and your battery will discharge fairly quickly.
Hope this helps.
Ken