Just an opinion...Reviewing your specs and what would seem to work - I think I might take a hard look at a Nikon Monarch 3, 4-16x50 or even the 42mm objective version. I own both and would go with the 50mm...a definite edge in low light whether its in timber or a dark shadowy draw, especially if you need enough magnification to evaluate a rack to be sure your target is legal.
Pros - pretty decent glass, providing acceptable clarity for the price. And if you look around with a little patience you can generally get one for about what you've budgeted, or at least reasonably close. The BDC models are more common but the Nikoplex reticle versions often include provisions for a custom calibrated elevation dial. Weight runs about 20 oz or so and the scope itself - 1" tube - is easy enough to mount on either a short or long action without spending a fortune for quality rings & bases. Also, from experience - holds up to recoil quite well, and the side-focus parallax adjustment does help for longer range work when used properly.
Cons - The Mon 3 line did not include scratch-resistant coatings so be aware of that when putting it through rough use. If you buy one new the included flip-up lens caps are pretty flimsy, so plan to use something else in the field. AND FYI - Definitely need to run your ballistics through a calculator if you use the BDC, as their ballpark guestimates are just that at best. Fortunately the Nikon SPOT ON app seems to give reasonable estimates out to 400, but beyond that...well, just say that environmental factors are always an influence so you might need a bit more detail from your choice of ballistic program. Either way I strongly recommend some range time to be sure you have an accurate chart for your hunting rig at the distances you actually intend to use it for.
Suggestion, just a thought - keep an eye out for a Monarch 5 while searching for a Mon 3. Yes it's going to be a little more money, but the ED glass does seem to take it up a notch for low light situations at longer-but-still-practical ranges. Does so without jumping all the way up to the prices you see for the higher-end Leupolds, Zeiss HD, Nightforce etc. That said, I haven't run a Mon 5 through a rough hunt yet to test it's durability, so if anyone has experience doing so please chime in!