Mueller APV is closer to your original budget, and is a nice scope. Not as nice as a VX-II, though. I would prefer a higher end magnification scope for that rifle if it was mine, like -12 to -14 or so. I like magnifying the tiny targets I use for 22lr more.
If you can find one of the closeout Bushnell 3200/4200 series in a scope you want, those are some nice glass. Bushnell cleaned up their product line recently, and brought them together under the Elite name, orphaning the 3200/4200.
Vortex makes an adjustable objective crossfire, 4-12x40 - about the same quality as the APV. Nice scope, not a great scope. Swift might have something that would work in their Premier line, too.
If it were me scoping that rifle I'd be looking for an adjustable objective, upper end magnification of 12 to 16 power, 40-44mm objective, and ideally a mil-dot reticle. Check out the optics section on the RimfireCentral.com forum, or do a search here on scope for a
10/22. That seems to come up every other week.
Swift, 17HMR Savage
Mueller APV, 22LR Savage