Yep, I have adapted Sony Zeiss 16-35mm f/2.8, samyang 14mm f/2.8 (a-mount version), canon 17-40mm f/4, nikon 14-24mm all with no issues on A7. I also personally know someone who used canon 17mm and 24mm TS-E on A7R just fine. I am pretty sure DSLR lenses with a good adapter works just fine (some adapters like cheap commlite are not good with UWA lenses)
These come to mind...
Samyang 14mm f/2.8 - DSLR version within budget, but massive
Samyang FE 14mm f/2.8 AF - native e-mount version with AF, but massive. Buy used or grey import to fit within your budget
Voigtlander 15mm f/4.5 - not your preferred speed but compact and if you buy used the m-mount version would be within your budget
Canon 17-40mm - if you bought used and with some patience could just fit within your budget (inc. mc-11 adapter). But once again huge.
Canon 20-35mm - old zoom lens with average performance but a budget option.
Voigtlander 20mm f3.5 color skopar - available in canon EF and nikon F mount. Doesn't have good reviews for sharpness. You could afford used one within your budget.
Sony/Minolta 20mm f/2.8 - a good sharp lens. Not overly big. Will AF with LA-EA4 adapter.
Canon EF 20mm f/2.8 - again you could buy used to fit within your budget. But bigger than the minolta/sony version.
Olympus OM 21mm f/3.5 - tiny lens, mostly nice and sharp. You could try and convince me to sell you mine