Getting the crown race seat properly milled is the best solution, but not all shops have JIS crown race mills. Failing than, careful filing to remove the chrome and other obstructions will likely succeed. Are you greasing the parts before trying to install them?

Using the old race is a possibility, if it is the same thickness as the new part. The downside is that the crown race bears the brunt of the abuse a headset suffers over the course of its life, and is the most likely reason to replace a headset in the first place. Why are you replacing the original headset? If there's obvious wear on the original crown race, you are unlikely to find any significant improvement with the new headset if you continue to use the old crown race.