Robert (born 1867 and from Scotland) appears on the 1901 Brighton census as you said, and can be found in the 1911 census, although he seems to have moved into the Steyning district (Hove or Preston). His wife and daughter were both called Christina, hence Chrissie.
Robert married Christina Sutherland in Brighton in the September quarter of 1891.
GRO ref 2b 410
A copy of that certificate could name his father or give some idea of his parentage.
It may be a false trail, but the 1881 census has a Robert Gibson born Scotland 1865, Tailor's Apprentice living in Cumberland with his widower father (a tailor & draper) and 8 siblings.
You also say you can't find a record of Robert's birth, I just have to ask, "You have been looking for it amongst Scottish records not the English?"
One census gives his birthplace as Crocketford Kirkcudbright, would that location have featured in Janet's history? It is not too far from Dumfries. Roy G