I don't have a definitive answer for you as my time is limited right now, but I think I have enough to give you a little overview of Romney. The most helpful item I found was this 1961 label for Romney Exclusive, which you can see is nearly identical to your label except for the store:
I also found references to Romney exclusive items in the 1920's. A 1928 ad featured
"Romney" Dresses and "Romney" Coats, which they noted were
"replicas of the most successful Paris creations." Throughout the 1930's and 1940's sundry stores noted that were
exclusive sellers of Romney fashions--which included coats, dresses, handbags, shoes, gloves, etc.
The two Romney-for-Godchaux's items I found were from 1962 and 1967, which underscores the 1960's suggestions you were given in the other replies.
Some of the items I found suggest that Romney items were a product of Romney Fashions Inc., but I did not find a record of the trademark owner.