I believe the problem in doing that conversion is the bolt head size. The rimless .35 Rem has a much smaller rim than the rimmed 30/30. You may be able to get a bolt from a .35 rem, and replace the one currently in your gun.
I believe there is a cartridge that's essentially a .35-30/30 that might work, also, the .32 special is a cartridge that is just a 30/30 shooting a .321 bullet. If you have access to re-boring, that might be the easiest conversion.
The .356 Win is a semi-rimmed cartridge that has the same rim diameter as the 30/30, but has ballistics very close to the .358 Win. Converting to the .356 or .35 Rem. might involve other modifications to the magazine and feed systems.