First off empty weight is useless and you will probably end up 500-1000# more than the empty weight. Also cargo includes passengers as well the dog and anything else you throw in the bed. Bottom line is you'll probably end up closer to 8,000# TT plus what you add to the truck. The tongue weight will be around 1,200# so you'll need probably 2,000# of truck cargo capacity. If you have a 10,000# GVWR on the F250 the diesel version will weigh around 7,500# so you'll have about a 2,500# cargo capacity.
Good luck and just verify the spec's of that individual truck before you buy, as not all F250's are created equal.
Ford Tow Guides
2012 Sundance 3100RB w/Reese Goose Box
2004 Ford F350 6.0 L PSD, CC, DRW, long bed, B&W drop ball hitch, Firestone Ride-rite air bags.