I think you are going about it the wrong way looking for a new rifle. First off what is wrong with your Mauser's barrel? I'm guessing it heats up too fast, and won't allow your 20 shots between breaks. Hate to break the news too you but I doubt any barrel on any rifle your looking at in a $500 price range will hold up to 20 shots without wandering. Even 1000 yard benchrest shooters only shoot 10 shot strings, and a sporter weight barrel is only good for around three shots.
I think Nathan has the best idea so far, possible new barrel and stock, or a scope that has repeatable adjustments.
If your Mauser 06 shoots pretty decent already I'd spend $500 on a new scope (possibly mounts), trigger, and bedding job. I'd look at something like the Nikon Buckmaster 4.5-14X40 PA scope for $300, Timney featherweight or sportsman trigger $50-75, skim bed and free float w/o pillars $100-150 or with pillars $200 at most gunsmiths. You might even look into a new firing pin by David Tubbs
to improve the lock time on an old Mauser.
If it is time to rebarrel that old Mauser call IT&D Customs
they have a great reputation for good work done fast at a reasonable price. Average turn around time lately has been 6-8 weeks and around $400 for a rebarrel with a Douglas barrel in contour and chamber of choice. However I'd probably stick with the 06 or .308 if you aren't reloading your own ammunition.
Lastly if you have to be able to shoot 20 rounds before a break you might try Teludyne Tech
for that. Have them add one of their straight jacket systems to you existing barrel on your Mauser. I don't know what the cost is, but they are supposed to dramatically reduce heat.