I think you might be able to find a used Vortex PST around that price, or at least close to it. Might be worth saving up another 100 or 150 to get solidly into that territory? For the price they are solid choices with great features for a precision rig. That viper listed above is a good choice as well and in your budget NOW and NEW.
The leupold patrol model is also around that price point. Slim and trim and a good fit for a handy little rifle. The SWFA SS optics are excellent, and the fixed power ones are generally very reliable AND solidly in your original price range. All the ones I have used are small enough to be handy, and most importantly had good reliable adjustments. A fixed 10 could fit the bill, or I believe they offer a 6x if you want less magnification.
Inside of 300 yards any magnification COULD be adequate, but I guess it depends on what you are shooting lol. I have a 3.5 ACOG on my carbine and it is plenty effective on steel torsos at distance, BUT it is not much of a target optic. My precision AR15 has an old slimline ergo USO 1.8-10 on it and I could not be happier. That magnification range lets me do anything I would ever want to do with that rifle. Something along those lines ought to fit well unless you are going full on target or bench style.
I would think something with 10x on the high end would do very nicely while not being too bulky.