Note: while I attempt to answer this question, I realize that this topic is extremely sensitive for many.
I apologize in advance to any that will feel offended by my words, it is definitely not my intention.
Just coming back now and realizing a crucial miswording in my question! My wife, children, and I left the church years ago.
Ok, since this has been cleared, I would have a fundamental frame challenge for you.
Losing my bedrock of faith, confronting my mortality, and losing the possibility of an afterlife with my own family was challenging and heart wrenching.
I cannot bear the discord of my children singing to a crowd a song extolling an afterlife that this religion teaches they can never have a part in because of their family's lack of belief.
Reading this it seems to me that you were (and are to this day) not convinced of your choice, and somehow you feel guilty for taking your kids away from the church.
Personally I would try to resolve this before going any further with your request of changing songs.
(I focus on you because you have not mentioned what your wife thinks/feels)
I would then pose to you the following:
if you (and your family) don't believe in an afterlife, you have lost nothing, as the lack of afterlife is for everyone.
if you (and your family) still believe in an afterlife, given the age of your kids, nothing prevents them from possibly joining again the church one day and be forgiven (and thus re-join that afterlife you consider "lost").
In any case, letting your children participate into the song would not be a problem:
if you (and your family) don't believe in an afterlife, the children are simply singing along some fairy tale (that the rest of your extended family believes to be very real)
if you (and your family) still believe in an afterlife, the children are singing about something you also believe (although possibly in a different way from the rest of your extended family)
In both cases, the rest of your extended family is happy, and your own family either has done nothing of consequence, or has participated in its own way.
What I am trying to say is that the problem does not seem to be the act of singing that specific song, but the meaning you give to it and the feelings this meaning evokes in you.
Your children are obviously sensitive to your feelings, particularly if you wear them on your sleeve, so it does not surprise me that
my children immediately said they hated the song - I think because they saw my reaction of distress upon hearing it.
I would then stress my frame challenge and invite you to reflect on your feelings, the meaning that you give to specific acts.
As an agnostic/atheist (depends on what definition you use) that also comes from a religious family and upbringing, I am sympathetic to your difficulties in shedding the baggage that comes with that, and that is why I am giving you this answer.
My kids are 4 and 6. They also receive comments from neighbors children, "I'm going to heaven, but you won't get to be with your family when you die because you don't believe." These comments are hurtful to my children.
This is a different issue, but important nevertheless, as it gives background to the question.
You seem to come from a deeply religious neighbourhood that does not tolerate secularism much, and I see how that can be difficult to navigate.
I would also have a couple of suggestions on how to alleviate the pain of your children in these scenarios, but as this is not the topic of the question, I will leave it for when someone will ask that question.
Nevertheless, this tells me that your community might be much more conservative than the ones I've been used to, so you might be going through more difficulties than me.
My answer would like to urge you to evaluate closely if some of these difficulties are self-inflicted, and could be removed by some reflection.