According to page 4-59 of the online 2000 Silverado Owners Manual, the "Max. Trailer Wt." for a K-1500 (4x4) 3.73 is 7500#.
According to the online 2000
TRAILERING WITH GMC TRUCKS, the "Max. Trailer Weight" on page 8, for the K15xxx trucks with 3.73 ratio, varies from 7000-8300# depending on cab style and bed length.
The GMC brochure also gives a GCWR of 13,000# for the V5300 with a 3.73 axle.
Based on your cab and box, you can decide what MTW value you want to use.
The MTW number (let's assume it's 7500#) assumes a base model truck with driver and one passenger.
The combined weight of additional passengers, optional equipment, cargo, WDH would have to be subtracted from the 7500#.
Assuming combined extra weight of 500#, and based on pulling capacity, the maximum allowable loaded trailer weight might be limited to about 7000#.
Assuming a trailer weight of 7000# and tongue weight percentage of 13% would give a tongue weight of about 900#.
Published "payloads" for the 1500 trucks range from 1852# for 4x4, regular cab, 6.5' bed, 119" wheelbase to 1652# for 4x4, 4dr extended cab, 8' bed, 158" wheelbase.
Again, the combined weight of extra passengers, optional equipment, cargo, and WDH would have to be deducted from this value.
The remainder would be the "payload capacity" available for the vertical load imposed on the TV by the TT.
Assuming 500# of extra passengers, optional equipment, cargo, and WDH, would leave about 1150-1350# available for TT-imposed vertical load.
Since the WDH can transfer a load equal to about 20% of tongue weight to the TT's axles, the maximum allowable tongue weight (based on TV's GVWR) would be about 1400# to 1680#.
Based on the above assumptions, it appears the maximum loaded trailer weight would be controlled by the "Max. Trailer Wt." value rather than by the GVWR value.
As others have suggested, since you already have the TV, the best thing would be to load the TV approximately as it would be loaded for camping and get it weighed.
Don't forget to include about 100# for the weight of a WDH.
Then subtract the expected loaded weight of the TV from the GVWR.
The remainder (plus about 200# for load transferred to TT axles) is the estimated maximum allowable tongue weight.
Maximum TT weight, based on TV payload, would be equal to max tongue weight divided by 0.13.
Then subtract the expected loaded weight of the TV from the GCWR value of 13,000#.
The remainder is the estimated maximum allowable TT weight based on GCWR.
Then use the smaller of the two estimated maximum allowable TT weights.
Ron