Did your TLS hub not come with a BSA Axle (not to be confused with the shorter triumph item)? That should fit just fine...
No it didn't have an axle so i can measure all the dims and compare them to the original 8 inch full width BSA hub [actually I have two of those], and they are identical.
I fitted an A10 axle in there and that sticks out of the left hand end of the hub more than the axles on the other wheels, even though all the dims and axles are the same ???
Looks like the wife is staying here for a while longer bless her...