#1 You are going to get a LOT of derogatory responses here because its a half ton (grocery getter) and because its a "foreign made" (see my signature) truck.
#2 Your biggest issue will be available payload when loaded up with your family and whatever you load in your truck. Go weigh your truck with the kids and some gear in the truck. Once you know your loaded weight you can determine your available payload. Once you know your available payload, you will have a better idea what you can haul.
I have a 2010 doublecab with about 1000 lbs of available payload after the wife, toddler and newborn are loaded. Figure 75 lbs for the WDH, I have just over 900 lbs for tongue weight and gear.
I load everything in the trailer (have 2500 lbs available payload in the new trailer) and expect to have around 800 lbs tongue weight LOADED with the trailer.
If you can keep the numbers right, the Tundra is a GREAT vehicle to tow with. I have factory E rated tires and added the TRD sway bar. My truck is solid when towing, I don't get pushed or pulled around when towing. Braking has always been solid (east coast towing, I won't get to the Rockies with this truck or trailer) and handling is great.
