30-06 and 210 bergers

[Brock]

I am building a house and i have no money for a new barrel, i would like to know which powder would likely get me to or over 2700fps with a 22in tube. My rifle is a plain jane savage 110, it's ugly enough to scare the bark off a tree but it shoots good. I would like to take it out to 800yds max on whitetail and song dogs once i get good enough. Is this even doable with a 22" tube or should i look at a lighter bullet? Thank you for your time.