Welcome to JOF
I'm a little confused by some of your data. First, I have to assume that you meant a BHS model, not a BUS....., second, the 2,000lb tongue weight has to be an error.
Here are the specifications for a 2005 Jay Flight 31BHS (from Jayco web site):
Unloaded Vehicle Weight 6,935lbs
Dry Hitch Weight 960lbs
Gross Vehicle Weight 9,000lbs
Cargo Carrying Capacity 2,065lbs
Exterior Length 32'-6"
Exterior Height 134"
Bedroom Interior Height 81"
Fresh Water Capacity (gals) 46
Gray Wastewater Capacity (gals) 32
Toilet / Black Wastewater Capacity (gals) 32
"Assuming" you were towing at the TT's GVWR of 9,000lbs, a 12% to 15% loaded
tongue weight would be 1,170lbs to 1,350lbs......, no where near 2,000lbs!
If your 2005 Sierra is a 1500 model, you may not have enough "payload capacity" to accommodate the combined weights of the TV's passengers, cargo, and the loaded
tongue weight of the 31BHS TT.
Don't know enough about your TV to comment, but based on my TV/TT ('05' Eagle 278FBS, 6,915lb UVW, 9,000lb GVWR) combination with a CAT scaled "loaded" TT weight of 8,400lbs (no fluid in tanks), a 1,250lb loaded tongue weight, I wouldn't want to tow my TT with anything less than my 2500HD/6.0L gasser.
Please note, a WDH must be sized to meet the loaded weights of the TT question, and a WDH will not compensate for over-weight TV and/or TT conditions. Also, think in terms of loaded weights, be careful using dry and/or unloaded weights.
Just food for thought.