One of the older Onkyo tuners in the 4000 series would be a good recommendation for you.
And they cost far less than your budget and are technically every bit as good as some tuner selling for far more money.
They are only about four and a half inches high from my recollection. Some old Yamaha tuners in the "T" series are also very low profile performers and generally sound decent, as well.
There you go!
Richard Links
Berkeley, CA