Hi Colin
Proof of birth would not have been required in the way I think you mean. If they remained local, which looking at the earlier censuses, they look to have done, they would have have been known to the local parish committee (although obviously not their full history including parents' names).
Am I right in thinking that William and Beatrice were brother and sister and remained single?
If I have the right couple, they show with another brother and sister in previous censuses. You could try checking for death certs for other potential siblings to see what they say regarding their parents' names. The censuses I have been looking at:
1871:Beatrice Scott, pauper, b. Hobkirk
William Scott, brother, pauper, b. Hobkirk
Address: Woodfoot, Kirktown
1861:Helen Scott 71 b. Hobkirk
Bertie Scott 68, sister, b. Hobkirk
William Scott 62, brother, b. Hobkirk
Address: Stobs Woodfoot, Kirktown
1851:Robert Scott 57 b. Hobkirk
William Scott 52 brother, b. Hobkirk
Helen Scott 62, visitor, housekeeper b. Hobkirk
Address: Birkwoodfoot, Kirktown
1841:Robert Scott 45
William Scott 40
Helen Scott 50
Address: Birkwoodfoot, Kirktown
Not sure about Robert but certainly Helen lived past 1855 to be able to check her death cert.
Not sure if you have come across this site regarding the Hawick poorhouse
www.workhouses.org.uk/index.html?Hawick/Hawick.shtml There is a link on the page through to the Scottish Borders Archive & Local History Centre which you could also follow up with the see what records there may be.
Monica
Added: There are some gaps on marriages for the 18thC which might cause a problem. This is the batch code for marriages for Hobkirk with corresponding years M117902 for 1726-1733/1808-1819