Except it's not a "Gimlet". The cocktail was developed in an era and locale in which fresh limes were unavailable. "Rose's" was a familiar brand name for a old antiscorbutic product, containing lime juice, sugar and Preservative(s), readily available to barmen in the UK and pink parts of the globe.

Call your drink what you may (and it sounds "drinkable"), but Gimlet it's not....(just as the current "Martini" crop has naught to do with Martinis, within which folks may battle over the choice of Vermouths or the substitution of vodka, but with an onion instead of an olive it becomes a Gibson, and even that early explorer who tried Sherry instead of Vermouth was off the reservation).