Don't. Let them sort it out. Just be there.
The most frustrating thing, as a person who has had losses in life of family and friends, is other people's compulsion to try to force you to be happy or punish you for your grief. It also is clear they odn't care about you, they just care about your grief affecting them in a way they don't like.
Let people be sad. It's normal, it's healthy. Grief is normal and it can last for weeks or months or years.
What you are doing here is called catastrophizing, assuming their life will spiral into collapse over the loss of a pet. Stop it. It won't. They will be sad for a week or few, and they will move on. Stop trying to control the situation and control your partner.
This post isn't about you helping them it's about you having anxiety over their grief not being in your control.