A fine No. 7 barrel would be too valuable to use for a project like this. There are still those receivers in the grease waiting for barrels.
The No. 4 shank is 1"-14. As mentioned, the Remington 700 can be easily adapted. These are 1 1/16" - 16. Face off the counterbore, reduce the shank diameter, and cut the new threads. Adjust shank length. The breech can be brought back to almost touch the bolt face (the .303 barrel stopped about a sixteenth short, to allow for rimmed headspace). An extractor cut will be needed.
Try to find a barrel with a faster twist, unless you are OK with being limited to the lighter bullets.
There is also the option of starting with a new barrel blank. This will be much more expensive, but you will be able to choose the contour and rifling twist.