Sheldon, thinking about the same thing right now. My conclusion is that I would look at a 243based cartridge. Great case design and match quality available if desired.
The 6 Rem is also wonderful and would be a great choice in a shorter barrel, but since you will be using a long barrel, I would consider smaller cases. The 243 and 6mm AI are very similar and equally accurate.
Big thing I am hearing about the AI of either version is throat erosion. Both have fairly short lives. In 1000yd BR, there is big interest in the 6BR Norma and wildcats based on improved 22-250 cases.
These are able to get the 105 gr bullets to 3000fps with very little powder, need long barrels though. This reduces throat wear and barrel heat. If your rifles will not see extensive shooting then I would suggest going 243 as a min. If going to do lots, consider a 6BR or one of these 22-250 wildcats.
I am presently shooting a 6BR with 87gr Vmax at clays from 700 to 800yds. Lots of fun and makes wind doping critical. Am looking at something that can launch 105gr Amax for improved wind drift. Probably use a 243 in a 26" barrel and muzzle brake.