Specs for the trailer have the gross (dry plus carrying) at 10,500 lbs. Depending on load, tongue weight will be 12-15% of that, so 1260 to 1575 lbs. In this case the dry tongue weight is ~13.5% of the dry weight, so you may want to use that percentage for the loaded weight.
Anyway you look at it, 1200 lbs doesn't seem enough.
And (and I know you don't want to hear it) you may not have enough payload in your 2500 Diesel for the trailer. Check the sticker on your doorframe (if it's there, not sure on a 2005) for the trucks payload. You could be close to your limit with the tongue weight, hitch weight, kids and some gear.
2010 Cougar 30RKS
2015 GMC Sierra Max Trailer
"Drinks for 6, Dinner for 4, Sleeps 2"