The .223 was originally a target round and I think based on the .222. The .22-250 is faster but unneccesary at 100 yards. The .223 is widely available in good rifles. I recommend a heavy barrel bolt action, but ar15 are very popular. I would go with that in the best scope and rifle you can afford if you want better wind performance than a rimfire. There are lots of rounds that are less popular but good for 100 yard shooting.
But I would consider a high quality target rifle in .22lr. This can range from an
international smallbore target quality rifle as used in olympics, to target model
Ruger or savage. Look at older model 22 target rifles too. I shot rifles used in biathalon competion as a trainer, and they are very accurate at that range. Plus you can use the olympic target stance offhand.
The .22 mag in a decent rifle would be fun and challenging too. I think a high power may get boring and unpleasant at that range. But there are lots of alternatives. Get a good spotting scope.