Just to give you an idea. This is what my shetlands can pull easily. These are all working on the assumption of competition fit ponies. All these drives are in Northumberland therefore think very hilly
Single pony (9.3hh) can pull two adults and fairly light vehicle round a driving trial can win it against horses (no extra time allowance). The weight of the adults would equal about 19 stone (two approx nine and a half stone adults) then the vehicle weight on top of that.
Single pony (9.3hh) can pull two adults (9.5 stone and 14.5 stone) with same vehicle round the lanes at trot for 14 miles or more easily without a break.
Pair of ponies (9.3hh) can pull the two 9.5 stone adults for 25 miles all at trot with only one half-hour break. With a fairly lightweight marathon vehicle.
Pair of ponies (9.3hh) can pull four people (all fairly light, approx. 1 x 10 stone 2 x 9.5 stone, 1 x 8.5stone) plus the marathon vehicle at trot for approx 10 miles (I'm sure they would be fine for further but have never tried).
Your pony sounds ideal....I wouldn't worry about anything bigger so long as he is fit enough for the distance you want him to do and the vehicle is suitable he should pull the family fine. Especially as the kids are little. I would get out there and have fun....and remember to wear hats just as you would for riding, if you don't ..at least put them on the kids.
If you are looking at governess carts - remember they are the most unsafe carriage available. If anything happens they are the only carriage which you can't bail out of so do bear that in mind. If you do look at a cob then a waggonette makes a nice safe alternative to a governess cart.
There are loads of vehicles available for a 12.2hh. It really depends on your finances and preferences. I would suggest looking for something pretty low and stable and light-weight. They can range from little wagonettes to exercise vehicles to marathon vehicles to gigs...you name it you can buy it...
Remember the vehicle weighs lighter once it is rolling than it weighs literally. A vehicle with carriage wheels (metal/wood) is heavier but will roll better on the road and so pull lighter than one with motorbike bike tyres. The bigger the wheels the easier the vehicle rolls - eg a high gig will pull lighter than a low exercise vehicle...but a low vehicle will be more stable and secure. A two-wheeler is more stable than a four wheeler (despite appearances!)
As for harness.....well mine all wear breastcollars for convenience (I can chop and change harnesses and it doesn't matter if they lose/gain weight). If the vehicle has a no swingle-tree then he must wear a full collar as a breast collar could rub badly. There is also the French Collar - a cross between a full collar and breast collar, quite popular in driving trials at the moment.
Anway I hope that is readable as I am rambling...as you can tell it is a big subject but I hope this helps a bit. Feel free to pm me.