I have a silver birch about 6-8 feet tall with multiple branches from a short trunk and we put it in a large pot for two years

I would expect quite a way. the only experience I've had is when I had a forest pansy in a pot on the border. the main root found its way out and over 2ft down. I had to cut it and the other roots in the pot took over the main job for the plant.

There should be plenty of roots for it to re establish.

I would think that, if your birch has produced any roots outside the pot at all, they could be extensive. In the back corner of our garden is a mature birch. Outwith the garden is a farmers field that has been in set-aside for about ten years. This winter the farmer took the land out of set-aside and ploughed it. The plough chopped up hugely thick roots from the tree and well out into the field, much more than I would have expected. I am not concerned for our tree because it will have even more roots beneath our own garden, but you need to be careful when you lift yours and dig well away from the trunk of the tree. btw, nice looking birch.

I wouldn't do it now, leave it until the dormant season, november to march.

I can't tell properly from the photo but which Betula is it do you know? We have B utilis which looks a similar shape and is growing much slower than B pendula