For your criteria, the Tikka T3x Lite Stainless is an excellent option. It's available in left-handed models in both 6.5 Creedmoor and .270 Win, has a 22.4-inch barrel, and falls right in your budget. The action is famously smooth and they are known for great out-of-the-box accuracy.

Another good choice to look for is the Browning X-Bolt Hunter Stainless. It also comes in left-handed versions and should be close to your price range.

You might also check the used market for a older Remington 700 Stainless, which can often be found at a good price. Good luck with your search.