I predict that you'll get variety of responses from "never in 1,000 years" to "sure, go ahead" which may not leave you any better off than you were before. Here's my opinion, having eaten one bird that was killed by a bobcat (saw the bobcat pounce on the bird, chased it away and the chicken died moments later) and one bird that was dying of a prolapse, I posted the same question and got the full range of responses.
Here's what led me to make the decision to eat those birds anyway.
1) I knew why they died...it wasn't like I found them staggaring around sick or recently dead.
2) Our ancestors regularly ate birds that had been shot with arrows or taken by a hunting falcon, caught in a snare and/or carried around in a hunting dog's mouth.
3) A generation ago, when you bought a chicken at the butcher, it was hanging headless, maybe plucked, but you had to do the rest of the butchering yourself...this applies to game birds too, they were often hung with innards intact for several days before being butchered.
4) You aren't planning on eating it raw, so if it was exposed to any additional bacteria from the bobcat (or in your case, dog), a thorough cooking should take care of that.
5) I did every step of the butchering myself, while the one killed by a bob cat was the first chicken I'd ever butchered, the other one was not, so I had a very good idea of what the finished product should look like, smell like etc.
Now, here's one thing different about my situation than yours that does concern me...the cooler. IF she's been soaking in dirty ice water all this time, I can imagine nastiness from her feet and feathers perhaps contaminating the meat. I'd be less worried if she were in a refridgerator or freezer, or even hanging in a cold garage. So, take that into consideration as you move forward with the rest of my reply.
So, my advice to you would be to butcher this bird as soon as possible and look for any damage to the innards that might suggest internal contamination from the dog's attack. If it looked like there was internal damage, I'd be reluctant to proceed because I'd be concerned that the meat had been contaminated by fecal matter or the contents of the gall bladder. If the innards looked intact and no other alarming signs were found, I'd probably brine the chicken, soaking it in salt water for several days. Because you didn't control the amount of trauma in the experience (I don't know if she was just pounced on or had been chased around the yard) her meat is likely to be tougher than it would otherwise. She probably wasn't thoroughly bled out either, so that will lead to tougher meat. I don't know how old she was either, but an older bird will have tougher meat so brining will help with all of those issues.
Once she has brined for at two to three days, I'd examine her again for discoloration or wierd smell and then proceed with cooking. Be sure she is cooked thoroughly, probably a long moist process like a crock pot, make sure it get's hot enough to fully kill any bacteria. (my husband will tell the story of being sent to intensive care about 30 years ago by an ex-girlfriend who slow-cooked a grocery store chicken all day in a crock pot but at such a low temperature that it was more of an incubator than a crock pot, resulting in a terrible case of food poisioning).
So, I'll probably be told I'm crazy for making this suggestion by at least one person who replies, but that's what I did because like you, I hate waste. Again, think about the condition of the cooler as well.
Good luck
