Thanks to all who responded. My boy is gone. I see him in the house, and hadn't realized how almost everything revolved around him at the end. I will be fine, though it will take a while.

When my dad died, I filled my step brothers coolers with beer, which I don’t drink. And anyone who wanted to say prayers or heaven, I instead told what a great guy he was and how happy I am to have had him in my life. I believe in pay it forward. Best way to handle it for me has always been get another fur baby and pay it forward. They want us to be happy.

Have a gathering to celebrate his life and journey with you Versions of all of his favorite foods and treats. Then tequila will be my next move

Don’t make a public announcement. Tell only the people closest to you.

Hiding our Atheism to avoid prEyers comes back to haunt us now that a death in the family is expected. .....Maybe a decade of Atheist Assertiveness would have trained people to STIFLE EDITH and quit selling religion for a beloved dog with terminal illness. ....you have 7 days to BOTH mourn a glorious 4 paw companion AND put believers on notice our LOSS is never reduced by hell threats nor heaven bribes and what religion includes human reunion post mortem with a dog already guarding THE PEARLY GATES ? I WISH I WAS THERE to comfort your furry pal and you upon euthanasia. ....GET A CLUE BELIEVERS we Atheists are not buying your religion 4 sale ....faith makes reality WORSE take your alleged gawd AND SHOVE IT

Take your best friend for a chocolate-covered chocolate donut (something normally forbidden!) on the day, you will both smile.
Religious people mostly do not regard animals as "worthy" of prayers, so the religious remarks, if any, will be directed at your well-being. Feel free to say "thank you", "it's not my thing", or ignore entirely, but there will be way less than you expect.

I'm sorry you've this heavy decision to make about your beloved friend.

While I'm not you, and you aren't me, being that if I decided to announce the passing of one of our dogs I'd expect no shortage of those who feel the need to express religious based sentiments. I can't begrudge them that unless they cross a line with something heinous like "god's will" or in a better place". It's hard enough to know what to say, even when you feel a friend's loss.
There are also atheist/agnostic , tho few, in my FB list who wouldn't say such things as prayers in a reply.

As someone suggested below, maybe best to share with those who know your thoughts so you don't have to endure the litany of "prayers" on your behalf.

Perhaps they're trying to give you a ''gift'' of comfort. In that case....the best way to shut them up is smile gently, nod and move on!

How about asking for remembrances in place of prayers?

I'm sorry for what you are going through it's never easy.

Also later when you aren't so sad - contact some local rescues and see if they would re home a pet for you - if you go first. Some places will do that. Adopt from them.

Then you still have a pet in life while you can.

First, let me tell you how sorry I am that you're going through this; I understand completely how difficult the process is.

I can't get too indignant, personally, about people expressing themselves the best way they know how. If they are sincere in their condolences, I appreciate it. The terminology of grief and consolation of same should be a heartfelt matter, and one's grief at a time like this, imho, ought to be focused around loving, and healing the hurt.

Kiss him between his eyes for me

I guess my first question is why are you "announcing" his death? Who outside of your immediate household that isn't aware, needs to be aware? Obviously everyone should mourn in their own way, but if your way includes broadcasting the news, you are setting yourself up for that which you seek to avoid.

BTW- My condolences for what you are about to go through.

Thank anyone who offers condolences and ignore the prayers.
And I'm sorry for your loss.