I do not own an Axis but I do shoot a couple different 7mm-08's. A bullet in the 120 to 140 range should shoot well. I load with Ramshot Big Game powder but several other powders will work great in it. For the most part mine seem to shoot most all bullet/powder combinations pretty well with the exception of the barnes bullets. The barnes however shoot well out of some of my other calibers.
Sierra makes a 140 HBPT in a game game king bullet that works well and I have shot nosler with good success.