Bazimars, good to be asking these questions first.
First thing to do is load up the family, fill up the gas tank and add the normal travel cargo to the truck and head down to the scales. Get the weight on each axle and a total weight (or add the axle weights) and add about 150# for a 5er hitch. This is you Laden Vehicle Weight or LVW.
On the drivers door jamb you will find a sticker with the GVWR or gross vehicle weight rating which is the most you vehicle is rated to weigh when fully loaded. Next find the rear GAWR or gross axle weight rating.
Look in the owners manual and find the GCWR or gross combined vehicle weight rating which the most your truck and any towed load combined weight is rated. This will be a number specific to the cab model, engine and rear axle ratio. If you do not know the rear axle ratio, there should be a axle code on the door tag which can be decoded by the dealer or possibly by a member here.
Now for the math:
GCWR - LVW = maximum loaded trailer weight you can tow.
GVWR - LVW = maximum loaded trailer pin weight you can carry.
Your pin weight plus the rear axle weight should not exceed the axle GAWR.
Now what does the trailer weigh loaded. You can ignore the brochure dry weight and brochure pin weight. This a base number and included no options. Any item listed in the brochure as an option is not included in the dry weight.
Inside the trailer (in a cabinet or closet) there is a tag listing the UVW or unloaded vehicle weight which should include any factory options, but none added by the dealer. Add to this weight, about 1/4 tank of water at 8.33 #/gal, and 6.0 #/gal for propane plus your foof and supplies. It is pretty easy to have a trailer at 750# or more over the unloaded or dry weight.
Pin weight can be estimated at 20& of the trailer GVWR for rough numbers.
The loaded weight and pin weight should not exceed the weights you calculated above.
The problem you will have with the 1/2 ton truck is the GVWR and the pin weight. You can pull a larger travel trailer than a 5er due to the lower hitch weight on a travel trailer.
More than likely, a 22' 5er plus the family will have you right at the limits if not a bit over the limits unless you load really light.
Have fun shopping.