Should be no problem to find someone willing to go all the way from KK to VTE. For the right price that is. Getting the car across to Lao should be no problem except for a 200 baht fee. I would worry about the dogs, but I guess that will be fine as well.

In case you don't find one, there is a 20 baht bus that will take you across the border, just arrive at a time when it is not busy, and you and the dogs will fit right in the bus

On the other side, drive a hard bargain, those taxi drivers would love to quote you upwards 800 baht, settle for 300-400, unless you live out in the sticks.

Oh and don't worry, you won't be walking across the border, you are not allowed to cross FSB1 by foot, you HAVE to take the bus or a car.