Hi
My advice is off the beaten path.
One of the most useful "Tone Control" in my opinion is a variable loudness. I know of two that are said to be efficient: Audyssey Dynamic EQ (DEQ) and the RME Variable Loudness. I have only experienced Audyssey DEQ and it is to me, mandatory.
If the Tone Control is to be used on an album or track basis, a two button, bass and treble, could be a solution. If, however it is used to modify the general sound to taste, I would suggest something more sophisticated such as an FIR PEQ. I could be called a fanboi, but an AVR compatible with Audyssey MultEQ-X App is the solution. Something like the Denon AVX-X3700H. You get 7 powerful amplifiers (I know, you only need 2), you get Audyssey Digital Room Correction, Dynamic EQ, and with the MultEQ-X App you get a very high and fine grained level of control on the room and sound correction. I believe you can implement up to 24 filters, I could be wrong, as it may support more...
Not the most efficient solution in term of electrical consumption and there is a lot of unneeded features but I don't know of any integrated that has this kind of audio processing power.. Perhaps the Lyngdorf TDA-1200. It is, new, above your budget.
I know, a bit against the grain, type of advice: Denon AVR-X3700H
In my opinion an AVR is the best anchor for any audio system music or HT, 2-Channel of Multichannel.
Peace.