I have a pile of recipes for 8x57I, as well as 8x57IR, 8x57IRS, 8x57IS, 8x60 and 8x60S. And I have tried a few of them.
What is your 8x57I rifle? Mine is an 1888 Commission Rifle, about as weak as they come.
What powder do you have on hand, or can get?
As Muir said, slug your bore. Just oil up a piece of 00 buckshot, which should be .320, and tap it into the muzzle with a wooden dowel. Then tap it out from the other end with a wooden dowel, and put the calipers on it.
Many 8x57I rifles were recut with a chasing broach to make them 8x57IS, because that is all that the .323 bore is - just deeper grooves. The .318 was wearing out too fast. Not all of them, especially military, have obvious reproofing marks.
The brass can be run with an 8x57IS die, and if it it is not tight enough, you can buy a smaller expanding ball ( at least on the Lee Dies ) for little money.
Remington 8x57 RN ammo is .321 inch and very mild, because they know someone might fire it in an older 8x57IS rifle. My 1888 likes it.
Hornady and others make soft 170-gr bullets in .321 inch for the .32-40 Winchester. You can load these up to 2,400 fps at 35,000 PSI and kill deer all day long.