I know, firsthand, about pet bereavement. When my doggie soulmate Louie started getting very sick at age 14-1/2, I knew his time on earth was limited. I asked my compassionate and wonderful veterinarian if his own dog was suffering like that, what he would do. He said, "I'd let him go. It's the merciful thing to do."
That's when the floodgates opened. Louie went over the Rainbow Bridge on a Friday afternoon. I cried all weekend. I cried pretty much the entire next week. I still cry sometimes because I miss him so much. And I know this is a common experience when someone loses a beloved pet.
You're showing compassion and caring. You're allowing employees to grieve. That's huge to them, and they'll appreciate it more than you can imagine. For this reason, a number of leading companies offer pet bereavement leave and the trend is growing.
To get you started on your plan, here's a sample policy I curated and edited from several that I found. It's ready for you to adapt for your company.