The difference between 7.62x54R and 7.62x53R is because two countries (Russia and Finland) rounded the odd case length differently; one rounded up and one rounded down. They are now different cartridges with slightly different official dimensions. The difference in case length, distance to the bottom of the shoulder, and distance to the top of the shoulder is 0.1-0.2mm less in the x53R version (.004" to .008"; thickness of one or two pieces of paper). If a factory spec 7.62x54R case wont fit you have an oddly tight chamber. The two cases are supposed to be interchangeable except for the bullet diameter; the x53R uses .308" or .309" bullets where as the x54R uses .310" to .311" bullets. I've never seen 7.62x53R dies before and when I looked into it a couple years ago you are just supposed to use 7.62x54R dies with a .308" expander ball for the neck (and .308" bullets of course). If a full-length resized 7.62x54R case doesn't fit your chamber something is odd.
What kind of ammo did you try that wouldn't let the bolt close? If it was surplus it could just be off. I'd try new production commercial ammo like Prvi or S&B. If those don't fit you'll have to handload.
Lapua does make headstamped brass and loaded ammo for 7.62x53R but Hirsch Precision doesn't have a listing for the loaded ammo; just the brass. Chances are factory ammo will be very expensive as well; probably $2-3/round. I'd handload personally, as mentioned above with 7.62x54R dies with a .308" expander ball, but if they don't chamber you'll have to modify something. Could try grinding down the shell holder so the die sizes the cases further; removing .004" or so wouldn't be too difficult with some wet sand paper on a piece of glass or something else flat. Shell holders are a lot cheaper to replace if you screw up so I'd avoid modifying the die.
Could colour a factory round with a marker and see where it's contacting. If the larger bullet is contacting because of a super short throat or something then it's not the brass.
If you're going the handload route I wouldn't bother with official 7.62x53R brass unless you really want Lapua precision or are obsessed with having correct headstamps. I'd just use any 7.62x54R brass and size it shorter if necessary and keep it in specifically marked boxes.