No, 3/8" thick is not OK!(0.375" is probably OK, but it would stick out 1/16")
The oil seal is inserted with the open end towards the bearings, and most of the stiffness is in the closed end, which would be out in fresh air. The seal will appear to be fine, but without the closed end being within the hub, the seal can very easily work its way out, and allow the grease to escape.
From a non-Minor application I have seen this happen, with a very similar over-width oil seal.