I don't think you register the truck for the weight it's towing. You register the truck for the weight it's axles are supporting.
Is this a GN or FW trailer? The rules might be different for that but I think the truck is only registered for it's GVWR. That's how all my trucks and trailers have been for over 20 years. But, I don't have any GN or FW trailers.
My dually (it's registered commercial) has a GVWR of 11,400 but I have it registered at 14,000. My 24' flatbed is registered for it's GVWR of 16,xxx pounds. When I have my dually loaded heavy and the trailer hitched up it sometimes puts the truck just over 13,000 which is why I registered it for 14,000. Thus, I maintain my Class A CDL from my truck driving days even though I'm an electrical contractor now.
