All No. 3 (F) rifles were Winchester based. Even if those were converted in Lithgow to snipers, they wouldn't be a Remington rifle. I therefore still see no reason why it would be a Remington rifle they picked.
Summarizing what you have is a Remington rifle with modifications in similar manner to what are required to have the PPCo scope bases fitted. These are basically a very small inletting on the receiver ears, and drilling two holes on the receiver. Nothing which can only be done in a military workshop and when considering that reproduction mounts are available it is quite possible that someone just wanted to have a sniper rifle built himself, but picked the wrong action and didn't complete his work.
There is no proof that this rifle was started by Lithgow (why wouldn't it be completed?), and even if they had started to work on it, why would they sell it off partially converted rather than throw away the action which is basically weakend.
Edit: does it have a D proof on the left side of the receiver? The AustralianNo. 3 T snipers I am aware of had those.