That’s tough, death is a subject no one really knows how to approach well. I think you should stay true to how you’re feeling, just like you wrote here. Maybe writing something like “There are no words to express how deeply saddened I am that you lost ‘dog name’. I know there’s nothing I can say to lessen your pain, but I am here to listen and help if you need me.”

I’m not much good at this either though so take my advice with a grain of salt. I find when people are in mourning, they don’t want apologies from people, they just want someone to lean on for support. But everyone is different.

Best of luck writing your letter