It should charge the same from either source as it's the charger that determines the rate.
What gauge wire is going from the house to the alternator? If smaller than 10 gauge, I would recommend re-wiring it. Then you can set it up so you can jumpstart the engine from the house batteries. It's easy to do and you can add breakers at either end. Biggest cost is the battery cable (you want 8 or 10 gauge). I did mine for under $100 total.
If the wire is already 10 or 8 gauge wire, then it would be cheap and easy to add the jumpstart capability.
Relays are cheap, you might want to try just replacing it. If it doesn't make a difference, return it.
