I agree with involving your kids in the decision, but I also think it's important that they understand that they can take time to grieve and be sad.
We went through 2 fish FAST and my oldest was all for going out and getting another. I said, "No, clearly we don't know what we are doing and we need to give ourselves time to get used to missing PaPu PaPu 2".
When our tortoise unexpectedly died (she was only 7 and should have lived to be 60+), we spent a good bit of time working on "how to grieve." When the cats or dog dies, I'll encourage my kids to honor what they are feeling then as well. It will be hard (for all of us) and there will be lots of comforting, but we'll go through the process.
The kids have all ready lost some loved ones (people) and the oldest is old enough (age 6) to have felt it (I think he was 4 the first time). I think dealing with the loss of pets helped him to let himself grieve and to honor that process.
While you should do what seems right for your family, I'd suggest giving the kids some time to BE upset and sad. This IS upsetting. It IS sad, and it's healthy for all of us to allow ourselves those emotions when they are appropriate.
Of course, we hate seeing our kids sad, and it's good to comfort them. But dealing with sadness is something we all need to learn how to do. If I felt I was mostly trying to cheer up my children (or me), I'd wait.
Anyway....I hope it all becomes moot. I hope Charlie is wilier than the coyotes and will be surprise you. Good luck to you.