1969lbs is pretty good for payload.
1969-625 (family) -200 (bikes) - 75 (hitch)= 1069lbs left over
I would keep at least 200lbs in reserve for a growing family and anything else added to the truck. So let's assume a maximum tongue weight of 850lbs for the trailer.
Most travel trailers do well with 13-15% tongue weight. So you should look for a trailer with a GVWR of around 7000lbs.
The 8250lbs of the 27BHS would have a tongue weight of around 1072lbs.
The heavier 9250lbs of the 28BHBE would be around 1200lbs. Both of those are using 13% tongue weight.
Even not loading those trailers up to GVWR you will be looking at over 7000lbs travelling weight.
The dry weight of the 28BHBE is 6510. So lets assume that is 6800lbs delivered with batteries and propane and any other options on the trailer. You will probably load an additional 1000lbs of stuff into the trailer. So lets assume 7800lbs loaded up and ready for the road.
7800*0.13=1014lbs of tongue weight.
Payload - family - bikes - hitch - tongue weight = unused payload
So based on those conservative assumptions you would have 55lbs of extra payload available. ~3% safety margin based on your available payload.
How much safety factor would you like while you are towing with your family?
Two other options...