Considering that your rifle is in very good condition I would be inclined to leave it as it is. There are plenty of MS in 30-06 that in average condition which are good candidates for a conversion and re-barrel.
You can open up the cartridge guide ring on a 30-06 a small amount to fit the 9.3x62 and it will feed flawlessly but a Dremel might be a bit rough, hard to get the finish parallel and square.

Post WW2 models are much easier to convert than pre-WW2 models as they were designed from the factory to be made in a multitude of cartridges with the minimum amount of changes, hence the Cylindrical follower and cartridge guide ring. On your model the diameter of the cylinder on the follower is designed for the size of the cartridge body and the cartridge guide ring is designed around the cartridge neck size.

Matt.