I must be bored, but I was thinking about this. I found
THIS pop-up that I would consider average for what you would be looking for. I also found
THIS hard sided trailer that is slightly larger but very similar in GVWR.
Keeping in mind what others have posted...the gas mileage really isn't going to be significant enough to matter on the average family trip.
They both have roughly the same GVWR 4500/4800. However, the popup is considerable lighter dry 2400/3700. This of course translates to more junk you can take, and with 3 kids you will need the essentials, plus some. The pop-up is also considerably lighter on the tongue 265/510 which could really make a difference if you want to put 5 bikes, 2 scooters, firewood and a kayak in the truck bed.
The PU will require mom to entertain the kids while dad sets up (yes, a sexist remark....maybe dad can watch the kids while mom cranks), whereas the TT can be backed-in already set up.
No more of that----that is the hole PU versus TT debate. We loved our PU, so glad I know longer need to set it up, tear it down, dry it out, etc....
IMHO....if you can do WITHOUT the clutter...hardside will have you camping faster and more comfortable. If you need the STUFF that 3 kids can be used to having. Get used to setting up/tearing down, cause you DAD need to sacrifice for that extra payload capacity. You can load 2000# of stuff in the PU only about 1100# in the TT, and don't forget that tongue weight.
EDIT: Just realized that the furnace and frig are optional on the pop-up and there is no mention of A/C. All of which come with the TT. There goes some of that payload. :S