One can never rule out the small possibility of such an item really being made. It is however very uncertain that they picked a very handful of Remington made rifles - I have never yet seen any convincing Remington based No. 3 T rifle. For me this would be necessary to at least give a slight chance there was another Remington made rifle which was - for whatever reason - stopped during conversion.

Another fact that speaks against this is the fact that I do not know a single No. 3 T rifle with Australianicon acceptance/proof/property marking which features anything particular Australian, aside of the marking. These rifles ALL carry the PPCo scope, the scope blocks are all shaped exactly the same way as the WWI scope blocks (in contrast to the 1935-1938 scope blocks on the BSA made rifles for Ireland! They are different both in shape of the bases, the way the serial number was added and if the serial number was with the W prefix or not). This for me is a sign that nearly all Australian No. 3 T snipers were simply leftover Britishicon WWI No. 3 T snipers.

I do know some No. 3 T rifles which for whatever reason surface in Italy which are rather oddly spread all over the right side of the rear sight protector marked with D (arrow) D and the rifle serial number, compared to the D with the broad arrow in it on the left side of the receiver towards the barrel as on my Australian No. 3 T rifle, which may make the "Italianicon rifles" a possible WWII Australian assembled snipers, but yet they are Winchester made too.

So we know that Winchester might not had been the one which allowed interchangeability of parts with the ones from Remington and Eddystone, and we also know for sure that Australia had British WWI No. 3 T's. Why then now start a manufacturer mix which can end in having no interchangeability between your own sniper rifles. These are highly valueable in war, even under great pressure you would very carefully produce them and make sure they are not lost or damaged, and to have spare parts available (fun fact from WWI: an Italian sniper always consisted of two persons - the sniper and the person who was in charge to bring home the rifle in case the sniper was killed, because it was so valueable).

I therefore understand your point and I can't rule out 100% that there was never such a rifle as you mentioned. What I tried was to give you facts which speak against this. If you want to believe in the 1 in 1 Million chance it's up to you, but always remember to buy the gun and not the story. And even if you had been the engineer at Lithgow who has drilled and modified the Remington P.14 action for sniper useage you don't have a written proof of what this rifle is, just a chance of appearing more reliable than any gun owner who might tell you the greatest stories just to get a few bucks more for his rifle, or even just envies!