M-S system
Yes, I can think of a very big one. The magazine length. A 30-06 length case is much longer than the 9x56 case, and probably will not feed properly through the magazine. Also, the feed rails would have to be altered slightly. If you compare the cases you have marked "30-06", you will find out that they are much shorter than a factory 30-06. Someone MADE 9x56 brass out of 30-06 cases. If the brass cases are the same length, you do NOT have a 9x56 chamber, and someone has altered the chamber by boring it out. (This is the least likely scenario, because of the difference in length of the two Original Factory cartridges.
Basic available cases are used by people for cartridge conversions, where they take a commonly available case (the 30-06 for example) and shorten it to make other hard to get calibres. I have a book called "Cartridge Conversions". It is about 2 inches thick, and covers almost every cartridge ever made, and tells how to make cases for obsolete and long forgotten calibres along with modern ones.
You might consider the .358 Winchester. Although slightly shorter at 51 mm long, it would work better than the .35 Whelan.
But then again, why change the Calibre at all? The 9x56 uses the same diameter bullets so the barrel would be no problem. Reloading dies are available, and functioning would be the same in the rifle. Or, you might even consider the 8x57 Mauser. North American factory loads are anemic, but check out the specs on European loadings.