Smart move looking into your TV weight limits when purchasing a travel trailer, and as mentioned your TV's GVWR is an important weight limit to watch.
Also, keep in mind that a TV's published Tow Rating is based on the TV's curb weight......, This JOF sticky explains in a little more detail: http://www.jaycoowners.com/showthread.php?t=3866
As member "mrnoyb" stated, the "loaded" TT tongue weight is also deducted from the TV's payload/cargo capacity and GVWR......, a factor that many easily over look. I would figure the loaded tongue weight to be 13% to 15% of the loaded TT weight. Oh, don't forget the weight of the WDH as well.
Published TT Dry weights of most TT's aren't shipped weights..., I would add about 250lbs to round-up to a shipped weight.
IMO the TV/TT combination based on the weights you presented is doable, but in the same breath I would say that it is also a combination that one could easily exceed one or more specified weight limits......., all depends on one's loading habits with three kids.
Good to hear that you plan to weigh the TV/TT combination under loaded conditions...., the following link may be helpful when you get to the CAT scale: http://www.jaycoowners.com/showthread.php?t=3871