It is a bit interesting for me to read what others think of the various milsurps and their various strengths and weaknesses. Many excellent points have been made.
I reread the OP and thought what would be imho a) historical b) useful to hunt c) right at $500 or less.
Unfortunately some worthy contenders dropped out--one of my favorites is the Swedish 94 Mauser Carbine which is simply too high now with stubby sights (but diopters fix that), the 1917 US Enfield and its P14 cousin--too heavy in the field, the 1903a3 again over the budget, the k98 too high and sights aren't great as issued, Swiss rifles such as the K31 and the like are priced right but for hunting ammo you would have to reload for good performance. Mosins, whether Finnish or Russian, long and heavy in the rifle for hunting on the move and not pleasant firing in various carbine models,
Finally the Mk 1, No. 3 or the Mk 4, No. 1. For me, I would recommend based on what the poster said would be the Mk 4, No. 1--good quality priced right, with micrometer aperature sights, .303 is readily available, relatively low recoil, can handle most anything on the American continent, and pretty decent hunting ammunition is made for it. The rifle is not too heavy, the magazine is also a nice feature if you want to unload or if you want to buy a restricted magazine just for hunting. Last, but not least, Savage made some Lend Lease US Property marked Mk 4's that are pretty good as well as the Canadian Long Branch ones.
A runnerup would be a sporterized Krag carbine but it is getting difficult to get hunting ammo now unlike the .303. My personal favorite as far as slick action and good accuracy is the 1903 barrel and front sight retrofitted on a Krag receiver.
BTW, I owned many of the fine rifles mentioned and handled/fired most of the rest above before losing mine in a tragic boating accident which sent them to a watery grave. I was saddened to see no love above for Arisakas nor the French MAS/Lebels or Norwegian Krags.
Good luck OP--hope your first purchase is not addictive.