I'm not familiar with the K-31's, other than what I've heard. Great accuracy, finely made, etc....
I don't know how consistent their bores are, so this is what I'd do...
Slug the bore at least twice, and compare measurements to make sure you have an accurate dimension for your bore size.
I'd then find a bullet manufacturer that can get you some bullets that are at least .001 larger than your bore measurement, .002 is probably better. Depending on the velocities you are wanting to achieve, make sure it's a gas-checked bullet. If you are going to keep velocities down below about 1800 fps, you can probably get by with a plain base bullet.
I would also clean the bore THOROUGHLY, to make sure all the copper and gilding metal fouling is removed before starting shooting the cast bullets.
Depending how the bore looks after removing all the copper, I'd consider lapping the barrel before shooting strictly cast. The smoother the bore is, the better luck you are likely to have with lead.
I'd start off with some moderate loads... see what kind of accuracy you get. USUALLY you get your best accuracy well below "max" loads.
good luck... hope it works out.. don't be afraid to experiment with different powders, and even bullets. If you can buy them 100 at a time to find out which style and weight your rifle likes, that would be a big plus, before you buy them in quantity.
If all else fails..... start casting your own!