I do not doubt that at least token purchases would have been made of the 1910, at the very least for evaluation purposes.
It is common knowledge that the 1905-model .32 WSL cartridge served as the prototype for the .30 M-1 Carbine round and that 1907 .351 WSL rifles were used by US the military for tactical-development and evaluation purposes in the 1920s and, by the French, as combat rifles during the Great War. This model, at very least, saw a great deal of law-enforcement use. NO reason why the more-powerful 1910 would not have been considered and bought likewise.
A part of the problem is that while some Winchesters have been treatised- and studied to death, others have not..... and this includes the sleek semi-autos.
The actual records of production should be in the Winchester Museum which is now amalgamated with the Buffalo Bill Museum in Cody, Wyoming. For something THIS unusual, they might even forego the regular charge for a Factory Letter which details the precise configuration of the rifle at shipping.... and to whom and where it was shipped.
There were certainly enough of the critters purchased by various Law Enforcement bodies.
I think that most of us are waiting with 'bated breath for some photos of this critter.
Any strange markings? Flaming Bomb? Government markings? Unit marks?
The more we know, the more guesses we can make.... and th better-educated those guesses will be.