I have no clue about the real origin of this phrase, but I doubt the connection with Burton the Tailors.
1. I have actually had suits made by Burton: I have never heard of such a suit being called "a Burton," and it seems not to be idiomatic for the English of the time. *"A Burton's" might be possible, but I've never heard it.
2. The connection with demob suits is just too late for the origin of the phrase.
3. Does it fit with the demographics of the RAF? Burton's were definitely one up from The Fifty Shilling Tailor, but I'm not sure they were officer class.
The explanation I've heard is the Burton ale one, but I don't put much faith in that, either. Sometimes the origin of words and phrases is just lost. How about the origin of "gremlin"?