This has been entirely unravelled by the fact that a UK birth certificate for Paul Edmund does not exist. I've searched for every time a 'Paul Edmund' appeared in a census record, and it does appear that in the earliest one (1901) he was boarding in a boy's home with Auguste's brother Edmund (listed as Edward) who had the unique middle name of 'Stiebel'. Paul Edmund randomly claims this middle name in the next census (1911) where he also starts claiming to be from Walsall (spelt 'Warsaw') in Birmingham instead of from Chelsea, London as he did in the 1901 census.
There is a 1919 shipping manifest on Family Search for Edward S Vargues. Address in England is his
brother P.E. Vargues, Uplands, Coventry
So cannot have ‘taken his identity’