Your 12v charger will nicely charge your two 6v batteries that are connected in series. Having your converter attached should not present a problem (it can remain attached when you do not have shore power). The charger won't elevate your battery voltage enough to risk damage to the converter; in my opinion.
However, the desulphating mode might concern me more. This is usually a higher voltage "spike" at a high frequency. I have no idea what this would do to your converter - likely nothing but it would be a concern.
I use a desulphating trickle charger on my RV during the winter. I hook it up to the two 6v batteries and let it run for weeks when it is quite cold outside. This has an automatic desulphating mode that always kicks in after the batteries get fully charged. This has not caused any problem for my converter.
Hope this helps....lets see what others have to say.