Children handle grief sometimes far better than we adults do. What are your plans for your dog's remains? We buried one cat ourselves (he died late at night and I didn't want to "store" him and upset the kids (very young). Two other pets, we arranged for the vet to bury them on his farm. But any way you do it, allow them to see the dog one last time and explain death if they are old enough to really grasp it. It broke my heart to have my toddler keep asking me every time we drove by the vet's office if we were going to pick up the cat this time or not. Ugh.
It's OK to let your kid see you grieve. Most times, a child's first experience with grief is the loss of a pet. It helps them be able to process harder losses that are sure to come as they get older --- grandparents, for instance. Let your child know it's ok to cry or not. Or cry together or cry alone.
I am very sorry for your impending loss and wish you the best as you make this very hard decision.
