Many of those 88/05 rifles are still .318 groove dia bbls (8x57I or more commonly called 'J'),,
the chamber neck having been relieved for use with later .323 bullet dia 8x57S ammunition.
That's why the fired case can easily take a .323 bullet,,,but the bore you are pushing it through may be .318.
Jacketed bullets can give good accuracy thru the combination.
Lead bullets generally not,,the ,323 lead bullet fired through a ,318 bore is usually stripped through the rifling and gives little or no accuracy. Alloy, speed, powders can all make some difference though of course.

This may be the problem and slugging the bore would be the first thing I'd do to see what you are dealing with.

If it's .318 bore,,you can size .323 lead bullets down thru a LEE sizer as mentioned.
If you want to shoot jacketed bullets,,I use .321 bullets intended for the 32Winchester Special (170gr) and load them lightly for the M88 sporters. The thinly jacketed 32Special bullets shoot well in the 8x57J bore rifles.
Just don't try and make a magnum out of any of them.