Since you are shooting shorter ranges I would agree with Robinhood and go with one of the smaller 6mm cartridges. I would also go with one that has factory made brass vs wildcats that need careful case forming. 6PPC or the 6mm Norma BR would be my choices.

If you don't mind a little forming (necking up) then the 6x45 would also be a good one as well. You could also use your existing .223 bold face and magazine without having any issues. Not as 'sexy' sounding when at the range talking about guns though :)

IMHO, the 6mm Rem, .243 and even the 6mmCM are a bit overkill for 300yd paper punching. If you were going to stretch out to 600yd then the 6CM would be a good one. Yes, the .243 is an excellent cartridge and if you were going to 1000yd I would consider it above the others. I would not like the barrel life but I don't want to buy a new barrel every year or two.