Sounds like your 15A connection has an open ground or something like that. You need to go get one of these 120VAC Circuit testers (LOWES) and make sure your house/garage 120VAC receptacle is wired correctly. You have to use all three connections.

Before you can plug into your 120VAC receptacle it has to show the correct lights on the tester. The bottom set should react CORRECT.
Then you need to plug in a good 10GA HD extension cord (The YELLOW ONES) that has all three conductors being used. You can use a 12GAUGE extension but if you start loading things down it may start getting hot on you. No way is a 14-16 WALMART RED COLOR Extension cord good for this use. You can eliminate the extension cord if you are close enough to the 30AMP RV Trailer plug. I use a 50-foot HD extension cord on my setup.

At the end of the HD 10GAUGE extension cord you need to pet one of these RV30A-15A ADAPTERS (WALMART HAS THEM). Then you can plug your 30A Shore Power Cable into that.

DO NOT USE the small BLACK RV30A-15A ADAPTERS as they will start getting hot on you in a short time... Need to use the RV30A-15A DOGBONE LONG connecter adapter.

Those that try to make short cuts here end up getting shocked by touching the trailer or worse. You trailer grounding comes from the shore power cable so it must be connected through the 120VAC Receptacle.
I also like to use one of these 120VAC PLUG in 120VAC VOLTMETERS. Mine is plugged into a receptacle inside the trailer in a good spot where I can glance at it. It has a GREEN SAFE ZONE scale on it and if the 120VAC power I am hooked to is outside this SAFE ZONE LIMITs I will not turn on the air conditioner for instance.

To answer your other question YES you can hook a 120VAC Battery Charger and connect to your trailer battery terminals. I carry the Black and Decker Smart Mode 40AMP SMART MODE charger model VEC1093DBD just for these occasions I need to charge my batteries and my on-board converter/charger may have failed.. Got to always have a PLAN B. I plug my smart mode portable charger into my generator and can re-charge my batteries to their 90% charge state in as little as three hours generator run time.
Just some of my comments here
Roy Ken