Why are you getting his cremains back ? I've never heard of this. when we euthanized our pets, we never saw them again. We got the collar and leash back. what I would do sounds callous, so please know I make a definate line between pets and humans. I would either throw the box away, or bury it, or find out if there's enough nutrients in what's left to mix them into the gardening soil and have it enrich your garden. I would not hang onto the cremains of my dog for ver long. What's the point ? Of course, we live on 8 acres, so we have plenty of space to bury our pets, but given where we live, they tend to get "un" buried by coyotes and eaten, no matter how good a job we think we've done. I just shrugged and told me kids, "Well, honey, I'm really sorry, but it is the food chain, and it does happen." (My kids are obviously older than yours)
I think the kids will have an easier time coping with this than Mom and Dad, frankly. And they will have an easier time coping if Mom and Dad just act as if it is a part of life. Pets come and go, hopefully the people stay.