Surgeguard doesn't make any claims to the accuracy of their shore power line current measurement. Since they only display current to the nearest amp, I would guess that the accuracy is no better than +/- 1 amp.
So you see 1 amp, but it could just as easily be suppling 2 amps. At 122 V, that could perhaps be as much as 17.8 A if we neglect the efficiency of the stock converter.
The stock converter won't charge your batteries as fast as a proper three stage battery charger. Since your converter simply applies a constant 13.7 V to the battery, the battery charge current will slowly decrease at the battery charges towards this target.
2015 30' Classic, "Chez Nu"
2014 RAM 2500 w/Cummins Diesel
ProPride Hitch, 400 W Solar