You will get a lot of data.....some good and some baloney...about the AI cartridges. Generally they're over rated for reasons that I don't understand.

I'd guess that a .250 AI with 22" barrel might be able to propel a 100 grain bullet at no more than 3,000'/sec and a 117 grainer at 2,800'/sec.

If you want .257 Roberts performance I'd suggest you build one and not try to milk it from the .250-3000 case.

My stock 700 Classic in 250 Sav, 24" bbl, will easily do 3000+fps with a 100gr, and near 2800fps with a 120gr, using Lyman data. Velocity from a 22" tube should be only moderately less.
Benefit of AI-ing a 250 Sav is you are still working with a case and OAL that shines in a short action, whereas the Roberts case gets a little iffy, especialyy with the longer bullets.
Would I change mine? No..but if I was building a rifle? Yes.
Major desirable factor, in an effecient package.