I'd use one of the SA made SMLE bbls personally.
The profile was used. I have 2 Lee sporters with that profile on them now and had another that I sold.

If a better bbl (heavier) comes along later like the early Metford you're wanting, you can always replace the lighter profile SMLE with the Metford with some stock work.

The Lithgow bbl would make a nice bbl for the project, but collectors and restoration folks would cry most likely.
Sell it to them for the big money and buy a custom bbl perhaps for the project(?)

It's been my experience that SMLE bbls fit just fine in the Long Lee actions,,the threads time the same. The only thing I ever did was to make the extractor cut into the radius cut of the Long Lee type bbl from the flat style cut of the SMLE.
Maybe that wasn't even needed, but it doesn't take much to do.

I have one of the Ishapore 410 SMLE bbls screwed onto one of the sporters currently. Works fine for a 2 shot repeater w/ 2 1/2" shells.
That one I had to lightly lap the threads into the action as they were a tad tight, but they indexed fine as usual.

Just some thoughts