OK/ You need to pull up your big girl pants and show your children how to handle this. It's an important part of owning a pet, being able to ensure they have a calm and peaceful end.
I have been through it many many times when the DC were little. Dogs, cats, rabbits, guinea pigs, precious precious rats.
I was very clear with them right from the start that this was us being kind and good owners by letting the much loved pet go without fear or pain. They had to hold it together, stroke them, tell them how lovely they were. Anyone who freaked out or did anything to distress the pet would have to leave the room. Weeping and wailing could - and did - take place once the pet was dead.
Reiterate to them that this is a time where they show ddog how much they love them by being the best friend they can, they have to stay calm. If they don't think they can - then that's fine, but they have to remove themselves.
My experience of endings has been 99% peaceful and not at all distressing for the children, apart from one rat (had to be DDs favourite) who struggled. All of the ddogs just went to sleep surrounded by people who loved them. None of the dc ever had to leave the room, but there were many tears once it was over.
Focus on ddog now and then the children would be my advice.