Wow, that is a really shitty situation. For what it’s worth, I’m with the people that think you should have been open with your kids about this from day 1, and then you wouldn’t be in this situation. But since you don’t have a time machine, the damage is done there.
I think telling her what you did about “I don’t believe, but it’s OK for you to believe and it’s OK for you to think I’m an idiot for not believing if you want” is probably the best way to go here.
Speaking from my own personal experience, I was raised as a devout Catholic and was a believer until I was 17 or so. My dad – who was divorced from my mom and who I only visited on weekends, so the situation is different in that respect – is an atheist, and was always fairly open about that with me. I went through a phase of praying for his immortal soul and thinking he was going to Hell, but eventually I grew out of that, and when I did, I had a lot of respect for him for always having been honest with me. My opinion of him would have been a lot lower if he’d made up some bullshit about being a Buddhist or let me think for years that he was a believer just like me or whatever. But that’s me, and my dad, and you and your daughter aren’t us of course. I’m just saying that that’s what my reaction was in a somewhat similar situation.
I haven’t actually been up-front with my own kids about my atheism, mostly because the topic just hasn’t come up, but I can tell you that after reading this thread, that’s going to change ASAP. We aren’t church-goers and my husband isn’t religious (side note, that thing about your wife saying “you should just be Catholic like the rest of and that would solve this problem” strikes me as being really shockingly offensive and something I’d have a hard time letting go, personally) so it’s less likely that my kids are going to run off in tears when they find out their mother is a heathen unbeliever, but still. No reason to hide my beliefs from them.