Prvi boxer primed ammo SP and FMJ made in Serbia uses copper jacket/lead core bullets so your good to go using that at the range,Prvi is also great brass for reloading as well as Wolf Gold line which is made by Prvi Win. white box 54r is actually S & B ammo with the Win. headstamp and there is also Norma and Lapua brass. As other have mentioned slug you bore before purchasing bullets so you get the best bullet to bore fit.
My M44 bore slugged out at .312 groove to groove,my rifle prefers the .311 150 gr. flat base jacketed bullets best,308 dia. bullets work but accuracy suffers as well as MV due to excessive blowby due to the smaller dia. bullet not filling the grooves as well. Depending on the range that your shooting at distance vs. accuracy may not be that much of an issue using a .308 dia. bullet so if you have some give a few a try and see how they do.
Cast bullets are a good option if you want to shoot cheap and extend brass life. In general you will want your cast bullet to be .001 to .002" over your rifles groove dia. For short range indoor shooting cast lead pistol bullet work just fine as
donor mentioned the Lee .314 TL314 90 gr. SWC is a great little bullet if it fits your bore. Just cast,lube load and shoot no sizing needed.
If you don't cast you can substitute the the Hornady .314 dia. 90 SWC in it place,the bullet is already prelubed but I add a touch of my thinned down Lee Alox/JPW lube for good measure,probably not necessary but what the neck. Accuracy is excellent out to 50 yds. and the 90 gr. SWC make an excellent plinker/small game load,MV runs around 950 fps. MAX using 3.2 grs. of Bullseye with no filler or is it needed you can increase it up to 5.0 grs. if you like a little faster load.
The same bullet also works well with the same load in X39 and about any 30 cal. rifle up to 06 case size including the 54r. If you want a little stouter load go with any of the Lee .312 dia. bullets from 155 gr. up to 185 gr. these all use gas checks and they can be applied easily with the Lee .314 push through sizer,these three Lee .312 molds tend to cast slightly larger than whats on the box all mine drop right at .314" and work well in my M44,SKS and 300 Sav. rifles. I push all these bullets with either 16.0 grs. of Alliant 2400 or 13.0 grs. of Alliant Red Dot no filler required. These are two standard loads by Ed Harris that work well in just about any military rifle.
How to slug your bore.
http://7.62x54r.net/MosinID/MosinSlug.htm
5 shots 50 yds.