The hard part here is the field of view. I don't know of any cameras outside of compact minidomes which offer a >120° angle of view, and they rarely include any I/O, let alone three. Even full size domes and box cameras are typically shipping with 1 in/1 out these days.
If you can't bend on any of these requirements, you'll likely need to use a fixed box camera with an add-on lens in an outdoor housing.
Panasonic's WV-SPN611/631 have three configurable I/O and two SD slots (edge storage was integrated with Genetec at least, last time I looked). You'd then be looking at very wide angle lens, something like the Theia SY125.
This is going to total something like $1,000-1,200, but it's the only way I know to meet the requirements you listed.
The sticking point is going to be the FoV. There are a ton of Bosch and Axis PTZ cameras with 4-7 inputs, but theyre going to be 70degress-ish at the widest FoV... HIK Vision DS-2DF5274-A has a ton of I/O, but a PTZ with narrow FoV...
Just noticed Ethans response