I found a source stating that there was no penalty for selling an un-proved barrel until the Gun Barrel Proof Act of 1855.

https://vlex.co.uk/vid/gun-barrel-proof-act-808271557

Easy to be an "expert" when there are no experts on pattern welded barrel identification, but IMHO those are English Stub Twist barrels. Unfortunately I have no list of early Birmingham tube makers to ID 'HSI'