H.F.
Get a cat or fish, anything that requires less attention. Or if you don't want to have any pets at all you could buy stuffed animals that look similar to your dog.
When daughter was 4 our cat ran away. We had had him for 7 years. He got out in the middle of a move. She was sad about it but not devastated.
Your children are pretty young and I think they will get over it quickly. Your 2 year old probably won't "really" remember the dog in a year or two.