Step 1. Buy a cheap, used 270 or 30-06 in whatever flavor of rifle you like.
Step 2. Send it to Jes in Oregon for a rebore
http://www.35caliber.com
Step 3. Enjoy your 338-06 or 35 Whelen in about 2 weeks after sending it in to Jes.
Alt: buy a cheap used 243 or 308, follow step 2-3 and enjoy your 338 Federal or 358 Win.
This can all be done well under 1k (+/- $500 for gun, $250 for rebore with shipping) all in and you won't be hindered by what rifles are currently available. You could also add a new stock and be just over that 1k mark, for example if you add a B&C Medallist for $280. Now you've got a nice semi-custom in a rifle you like in a chambering you wanted.
You could also go savage, and just swap the barrel. I see used Savage rifles for under $350 all over. There are dozens of options for a pre-fit barrel in whatever chamber you choose. Even midway and brownells carry them.