A little over your "hopeful" price, but a darned nice lens:
EF 16-35mm f4L IS USM (Note that it also has IS).
Older model that's not as good but is within your price range:
EF 17-40mm f4L USM.
How about a prime, well within your budget:
EF 20mm f2.8 USM (I use this lens quite a bit on both FF and crop cameras. Below examples are with it on 5D MkII)
Need wider? I've considered getting the
Rokinon/Samyang 14mm f2.8 (also sold under about half a dozen other brand names)... it gets lots of positive reviews and is very affordable... but of course it's fully manual.
Another lens that's recently caught my eye is the
Venus Laowa 15mm f4 Macro/Shift lens.... interesting idea and another reasonably affordable, fully manual (both focus and aperture).
I'd suggest you not get too hung up on an f2.8 aperture. Makes for bigger, heavier lenses, narrow range of focal lengths in zooms, and often aren't as sharp as slower lenses. f2.8 is pretty rarely a necessity with an ultrawide lens, too, especially when using a very high ISO capable camera. Ultrawides are hand -holdable at much slower shutter speeds, you'll probably find yourself stopping down for more depth of field most of the time, and there's little opportunity to blur backgrounds with an ultrawide unless you're working very, very close. A photojournalist or astrophotographer might want f2.8. But most of the rest of us can probably get by fine with f4 or even f5.6.
A little over your "hopeful" price, but ... (