My opinion would be to get a LA Savage already chambered in .270 or .30-06 so that you have the proper bolt face already. Then put an pre-chambered Shillen barrell in .270 Win. on it.
This would shoot more then well enough for what you're talking about. Would be a fun little project that would not break the bank, and you already have everything you need to reload for it!
Find one with an accutrigger if you can, glass bed it into a pre-finished laminate stock (not Richard's) and away you go! It's a win, win, win situation. That's what I would think.