"Rest in Peace"

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But notably chooses not to.

I like the Christian notion that people are born with, or come to (and move away from) a time in their lives where God is a significant potential reality
Liking something doesn't make it so. Its called 'wishful thinking'. The same thing applies to fairies.

I'm not saying it's true, even doctrinally, I'm just saying I like the idea and I think that sometimes people saying "rest in peace" is, for some people, the curtain flapping open...just a little.
For some people, that is almost certainly the case. I would even say that for most people, the desire for an afterlife is strong. But I must note that when such an occasion arises, they tend to gravitate towards whatever religion/belief is prevalent in their area, so it isn't so much that God is a potential reality for them but that an afterlife is a potential reality for them.

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To hear it, yes. To have it shoved down their throats. No. And the offensive behaviour was not at the funeral, but when the priest came to the house to give condolences.

If someone is secular, then get a secular interment. I find your statement here to be somewhat illogical and actually a bit naive.
There is a difference between an interment based on a particular religion and a deliberate attempt to get new converts due to the grief of the family.

Think about your neighbourhood JWs showing up at your door when you have a funeral and thinking that is a really good time to have a religious discussion.

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Well, it's a form of words. One wishes for something respectful to say and it isn't easy to say the right thing. Rest in Peace is accepted as the right thing to say so it provides the living with a way of respecting the dead.

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As I have said, a significant percentage of non-religious people either explicitly believe in an afterlife, or leave the possibility open. Those of us who do not may still use phrases like 'rest in peace' (I do not recall having done so myself, but I certainly could do so without feeling like I am being dishonest as I do with some other religious phrases) as a form of respect. To me, it fits well even with lack of belief in an afterlife. I know it isn't technically correct, but one doesn't get too technical with such things. 'Don't exist in peace' just doesn't have the same ring to it.

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I do not think you are lying, if that is what you are asking.

I think you have children; do you think them, or anyone else hearing you using spiritually emotive language will confuse how they observe your atheism?
I do try to be careful with spiritually emotive language, but as you know, I am a particularly pedantic. I do not find that particular phrase however to be religious in nature. Also, as I have said, I generally avoid mentioning my atheism at times of grieving as I find it disrespectful to do so.
As for other atheists, I do not know. I actually know very few openly atheist people outside of this forum.

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In this context, I would equate 'peace' with 'release' from whatever pain or suffering the individual may have experienced prior to death.

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Who said the phrase refers to the corpse? My own family members have been cremated, yet I think the phrase still applies. It is death that brings rest.

http://www.merriam-webster.com/dictionary/rest
Definition of rest
1: repose, sleep; specifically : a bodily state characterized by minimal functional and metabolic activities
2
a : freedom from activity or labor
b : a state of motionlessness or inactivity
c : the repose of death
3: a place for resting or lodging
4: peace of mind or spirit
5
a (1) : a rhythmic silence in music (2) : a character representing such a silence
b : a brief pause in reading
6: something used for support

at rest
1: resting or reposing especially in sleep or death
2: quiescent, motionless
3: free of anxieties

Note especially 'rest': 2 a. and 4, and 'at rest': 3.