I would call my local County Extension Agent if your area has such things, or Animal Control if not and ask their recommendations. "Indiscriminate" poisoning is forbidden in many areas for one. If it is a contagious condition the body may need to be disposed of more thoroughly than normal. This is a page from California that's a pretty good breakdown of things.
http://ipm.ucanr.edu/PMG/PESTNOTES/pn74123.html
I'm not a big fan of 'possums, but they generally coexist pretty safely and easily with humans, sure you don't want to be bit by anything. Like rodents, I think the problems are more things like ticks and fleas and with the droppings than other forms of transmission, but then I don't get that close to them.
Most 'possums understand cat very well -- if you hiss at them they generally will scurry off. I have some tall fir trees along the back of my property and from time to time they will seek refuge there, but once the dogs come out they usually keep on going. My dog Rio would have been so ashamed of my current dogs, she would never let vermin in her yard and now I think we have a bunny living under the shed.