Gentlemen,
IMHO, this is a no-go. I just miked some once-fired W-W 7x57 cases that were range pickups. I also miked a once-fired W-W 7mm-08 that was fired in my Model 70 compact chambered for that cartridge.
At a point 1.560 inches forward of the case head, the 7mm-08 case miked .456". The 7x57 case, miked the same distance from the case head, miked only .435".
A 7x57 Ackley Improved may indeed clean up the chamber but I don't have one of those from which to obtain the dimensions.
If I wanted more performance than the 7mm-08, I'd buy a Handi in .280 Remington. In fact, I just did. The increased case capacity of the .280, coupled with the 26" barrel that H&R uses, ought to give velocity approaching the 7mm Remington magnum.