I'm so sorry. I hope your dog's last days are super sweet.
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We recently had to put our beloved cat down, and while my son was 2.5 for the event, and didn't have to be home with us, it was still really important to us to handle it well for him.Â
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First of all, there was a local vet willing to come to our home, which made the whole thing soooo much better than at an office. If you can find that service near you, it's really worth the extra money not to have your child experience all this AND be in a clinical setting, with no way to distract herself if she needs to disconnect. If you can, I really would. And by the way, we found her at 8pm the night before! Your local emergency vet clinic that's open 24/7 might have that information for you, and sometimes your private vet is willing to make a house call.
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Second of all, the experience of the animal is often much more peaceful than we imagine, so I don't think you have to worry about your child witnessing her beloved dog's pain or trauma. If she hangs nearby, she will see some fiddling with veins, maybe a catheter, so you can let her know that Dog will have a kind of a shot, and sometimes more than one, and then he'll get sooooo sleepy, and then he'll be really asleep. Usually it's fine to keep petting and loving your animal up close during this so you could also direct her toward snuggling close by his face or brushing his back and she might then miss the whole shot thing.
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We chose to say, in a very very matter-of-fact tone, that "Cat has been having such a hard time being sick that she can't even enjoy her food now - and you know how she loves to eat. So she told us that she doesn't want to be in her body anymore. The way she loves us keeps going, and the way we love her keeps going, but the part of her that is a sick body needs to stop now, and that's called dying. She told us she wants to do that today." We said it like "Today we'll probably clean the shed while you are at school." (and in fact our hearts were breaking, of course.)
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Being a toddler, of course, he didn't even seem to have heard this, and he went off cheerfully to school. And, being a sensitive little dear, just after we'd finished letting Cat go, we got a call from school that he was having a really hard day, and they knew we'd rather not stress him. So we brought him home, and en route he asked "Is she dead yet? Is she still there?" So we did the explanation again.
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As soon as he came in the door he wanted to see her, and we let him pet her body and check her out until it seemed a little invasive (he wanted to touch her eyeballs...) Then it was clear that he was sucking up too much etheric pathos (while we tried to stay very matter-of-fact) so we moved on to usual activities. Later, he and I played in the near distance while his daddy dug a grave. Now usually, daddy + tool = total focus, but in this case I think he really needed the space. He didn't go anywhere near, but happily picked weeds/ flowers for the fun of it, and later was content to arrange them on the grave without discussion.
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So in all of this what worked well for us, spiritually, and as parents of this particular and super sensitive little boy, was to describe the process as only affecting Cat's body, and to keep him moving with plenty of ways to distance himself as HE needed, and with help when we saw him needing it. We still refer to the things we loved about living with her, and we playact being her just as we used to when she was alive. And we acknowledge missing her, and we have that moment with sincerity, but given who he is, it's important for us not to dwell on that or any sadness.
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Your daughter sounds very grounded, and loving. Perhaps some version of this?
"You take such good care of My Dog, the way you give him medicine for his owie. Unfortunately, his medicine doesn't work anymore. The next way we're going to take good care of him is by helping him die, so he doesn't have to have any more owies. I'm going to pet him quietly while he's dying. You could pet him also, or you could draw him a picture, or even just be in the carrier on me while you watch."
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My very very best wishes for you to find the setting and the rhythm and the tone for this passing that is best for all of you! And especially I'm sending snuffly doggy comfort for him.