I have no problem saying "Rest in Peace". I won't say that 'I'll send prayers', but I will say 'your in my thoughts'. I'll offer my condolences and respect.
I also say gesundheit or bless you (though I do feel silly) when someone sneezes. And I wish people a happy St. Patrick's Day and Merry Christmas, even though I don't recognize 'Saints' or 'Christ'.
I have no issues with the simple things that express some sort of concern or well-wishes to someone else.
It's a little difficult to navigate the word as an atheist when Christianity is so engrained into societal norms.

You got no fucking choice but to rest in pease because your dead

What if by some stretch of the imagination, the energy that leaves a person's body when they die was constricted somehow? Where does the energy come from that creates a deformed fetus? Aren't storms created from colliding energies? So couldn't a person say with uncertainty to the deceased...RIP? There is certainly...uncertainty in the world at every level!

Absolutely not. It's a prayer. And the alternative to resting in peace is burning in hell (like hawking is now according to the xtians). I'm not part of that.

Just what we need, the PROPER PHRASE POLICE........They're dead, say something comforting to the family ferpetssake.

I think resting in peace is fine, especially if your life was full of suffering at the end. Cancer, Alzheimers, or other health problems, death is personal peace, an end to suffering. Does not have to have anything to do with what you believe comes after.

You can say RIP instead of the full phrase. That way, in their mind it's "Rest in peace" and in your mind " it's rot in peace," I say this tongue in cheek, because I feel bad about any loss to others and wouldn't really feel that way.

I don't see it as pandering to false hope. I see it as acknowledging that life before death isn't 2nd essarily peaceful, so we want our loved ones to have whatever peace can be found in their afterlife.

Yes. Dirt naps are the most restful things ever.

My go to response to someone dying is "So sorry for your loss" and/or "My deepest Condolences"

I would say sorry for your loss amd wouldnt overthink it people expect you to say something comforting & I don't mind doing that because its ritual time and the whole ending of life is ritualised.

Regardless of belief it's been a method of connecting to people who were loved. We all liked actors and musical artists that have passed and it has become a universal standard of remembrance. It's harmless so why put so much energy into dispelling it?

It's a tradition nothing more, and it saves a lot of agro at funerals

Nothingness I don't know how much more at peace you could be ...