There's some ambiguity in the sentence, and this seems to be causing the grammar checker to misparse it.
The intended parse is
It’s not our (lack of (answers to these problems)) that causes us such pain.
"lack of XXX" is a singular subject, so a singular verb is appropriate.
But I think it's parsing it as
It’s not our lack of (answers to these (...
I think the confusion is that the word processor is suggesting using 'causes' instead of 'cause' - and you are assuming that it is because it is adding the 's' to make it plural.
However, 's' can be added to show that it is a persistent condition NOW, rather than in the future or past.
He will run tomorrow.
He runs out the door now.
That horse runs a ...
I agree, it's not the answers (plural) that "cause" has to agree with, it's "the lack of answers...". You have A LACK OF something, so the singular verb form is correct.
It's not the lack of funds that causes me such pain, it's the lack of fun.
I'd like to see what the grammarian behind that grammar checker has to say.