I had a .224TTH barrel on my Kimber 84M for a few years. It was a 27", 1/8 twist, 3 groove Lilja. I shot it sparingly, got 3700 with the Swift 75 Scirocco/ 3750 with 64PP. It's still in good shape, but doubt its had more than 300rds through it. Very flat shooter. I used R22 exclusively. I now use another rifle in 22/250 AI (also 1/8) with the Hornady 68 Match for I want a big 22 to do. I am sure you have figured in the fact that barrel life will be short, but that could take you weeks or years, depending upon how you use it. I think a fast twist is the way to go with a hot .22, especially with the 64 and up bullet weights, they just seem to kill better. I believe the higher rotation is a factor. I know, with the 22/250 AI above, a solid hit on a PD makes them go high in the air, spinning to the right! ha. Even a 1/9 will owrk with your 22/30.06. BTW, isn't the .220 Hawk just a 22/30.06 itself?
Anyhow, good luck in your endeavor, I enjoyed my 224 TTH experience, but I wanted the 7mm08 bbl back on the Kimber, plus, the 22/250 AI shoots factory loads well ( it's how I fireform my brass too).