Sounds to me like a bad cell in the battery. The charger will read that as low voltage and try to charge the battery boiling out the other cells.
13.5 volts is a normal charging voltage for RV converters (anything from 12.8 to over 14 is normal depending on the charging cycle). The converter is likely fine, but the battery is definitely no good.
2013 Jayco Eagle 324BTS
2018 Surveyor 265RLDS
2016 GMC 2500HD CrewCab SLT 4x4 Duramax
2014 Flaggstaff VLite 30WTBSS (totaled)
2015 Surveyor 264RKS (sold)
2008 Surveyor SV-234 (sold)
2008 Nissan Titan Pro4X (sold)