First, how old is the battery? Unless kept on a charge of some type and the plates covered with electrolyte (fill with water only), the battery will die rather quickly. I was replacing the battery on an annual basis until I relearned that.
Second, check the 120V breakers and wiring to ensure that the converter is getting 120V input.
Third, check to see that the converter is actually producing 13-14v output and that the output fuse is not blown. Ensure that the polarity connection at the battery is not backwards.
Hope that this helps with your troubleshooting.
Bill & Anne + FBs Athena, Jada & Cupcake
Wilson III: 2015 Ram 2500 6.4L Gas 4x4 4.10 CC SRW
Gypsy II: 2013 Kodiak 279RBSL with table & chairs instead of a dinette