Getting it done for under $200 is kinda difficult, but there are scopes out there.
I have a 3-9x40mm Bushnell AR scope, for the $130 I have into it ($170 minus $40 rebate), it does what it should. I'm sure the 1-4x Bushnell AR is on par with that scope. I never have problems with over-magnification, and I will point out the Nikon Buckmaster 3-9x40mm BDC's are still around to be had for $150, and they're a better scope than the Bushnell AR's.
If I needed a scope in that price range, the Bushnell Banner or AR and Nikon Prostaff or Buckmasters are the 4 that get my money.
You might find that a red dot wouldn't be as blurry as your iron sights. Would be worth trying one at a shop or from a buddy to confirm. I really like the Burris FastFire III, you won't find one for under $200 though except for clearance sales. Might be able to find an FFII for that price, but the FFIII is worth the extra $50.