Try some .250s in the .300 magazine, see if they work. Be very careful if you are using live rounds. Cartridge base diameter is the same, as are breech face and extractor. Here's where there might be a catch - Savage 99 locking is a bit complicated. There are critical surfaces at the top of the bolt, lower tail of the bolt, the tip of the lever, and upper and lower surfaces of the receiver. If there is wear, the rear of the bolt drops a bit when its locked, and because the main locking surface is slightly angled, headspace increases. Older Savages have a barrel shank with square threads, so they are a bit trickier to work on than never ones with V threads. There are also two cuts in the breechface - for the extractor and the finger at the top of the bolt which must line up. If there is excess headspace, and if the barrel must be set back one turn, the barrel breech must be refitted, as well as the chamber recut. This would also shift the location of the forend screw stud closer to the breech. Similarly, if the barrel turns in past index, the shoulder and shank must be recut to set the barrel back. The barrel might index, or could pull up short of index; this would be the best case scenario. I do not know if older 99s have uniformly timed barrels.
No harm would be done if the .300 barrel could be removed without damage, and the .250 tried. Then a decision could be made whether the conversion is practical.