It sounds like people are trying to excuse what is obviously an issue with the box just because it is from Snap-On. You paid $10k and so it should be perfect! If you bought a $5k car and it didn't go in reverse without jumping around you would take it back to the vendor so why not with a more expensive tool box?
From your video I reckon that the locks are not tight enough fitting and as such they are allowing the castor arm to semi rotate away from the direction of travel thus placing a side load on the wheel which is then transferred to the torsion bar suspension which (due to torsion bar suspension not dealing well with side loading) is causing the box to, "Jump."
If the castor locks were of high enough tolerance then they would prevent the sideways rotation of the castor arm entirely (as they should do) and the wheel would happily rotate freely in forward or reverse directions.
If the suspension was of coil spring type the springs would probably absorb the loading and you would not notice so much.
Just my two pennys worth. I do think though that if you are paying $10k for a tool box, that tool box should have zero issues and excuses should not be tolerated. In this country (UK) the Sales of Goods Act states quite clearly that if the customer is not entirely satisfied with the product for reasons of build quality, reliability or its fitness for the purpose for which it was sold, then you have the right to return it for a full refund.
I know you're in the US but your dealer surely knew the job you did, the place you worked and the purpose for which you needed the box. If the box does not do the job which you bought it for to your satisfaction then I would be looking for my $10k back.