I use magnification on my 10/22 (Hawke Vantage 4X32mm AO) because I want more precision at 50 yards and in (1" groups or less is my consistent goal). I tried red dots on it in the past, but they just didn't yield the precision I wanted.
But my AR-15 wears a red dot. After messing around with a couple of super budget red dots (e.g., Feyachi for around $40), I finally went with the good stuff: a
Sig Sauer Romeo 5, gen 2. MSRP is $159, which is only $10 more than your max, but they can be had for a bit less from reputable dealers.
And if I wanted one on my 10/22, it'd be that one.
The dot has 8 different brightness settings and is very crisp (even with my mild astigmatism, that causes "star burst" patterns with cheap red dots). It's also "shake awake", so it turns on automatically when you touch it (although it can be turned off entirely for transportation or long term storage), but automatically turns off after 2 minutes. Uses a common battery and yields somewhere around 50,000 hours.
It comes with two mounts: one high (for the AR platform) and one low (which you'll want for a 10/22).
My biggest recommendation -- especially with the Romeo 5 (gen 1 or 2) -- is to
buy directly from Sig Sauer OR a very well known and reputable dealer (e.g., Palmetto State Armory) because the counterfeits are now plentiful. (Ask me how I know; I fell prey to one on my first try.)
Forget about buying one on Amazon (even if they claim to be the "Sig Sauer Store") or ebay. You may get lucky, and a "good deal", but it's just as likely you won't.