I suggested a short lever action or a "scout" type rifle in .308.
A good idea, but I'm not sure you'll find a new scout rifle for under $500. I"ve seen the Savage 10FCM for just a bit over $600, but nothing under $600.
On the other hand, if you abandon the ready-to-go new scout idea, there are plenty of nice .308s for under $500. The Ruger American for example, can be found for under $400, and a Remington SPS can be had for not much over $500.
You may not have considered this, but don't discount the idea of a surplus rifle like a Schmidt-Rubin K31, Yugoslavian Mauser... or even a new lever action .30-30.