Perhaps an unpopular opinion, but I have never heard of quoted GPAs for purposes of employment selection being some subset of a cumulative GPA. I have seen and participated in many cases of CV reviews of undergrads at 2 firms, and while this was several years ago, I never came across this distinction once. So, on the slim chance it is noted, it would almost certainly come off as a deception.
That said, the chance that your stated GPA will be compared to your documented GPA does not appear to be great. I have less visibility into the background check process, but from my limited experiences when internal HR teams were conducting their document checks (distinct from an external background check), they weren't checking for comparisons; rather, they were checking that minimum hurdles had been scaled. For example, if the role called for undergrads from Princeton University with GPAs of 3.25 of better, graduated in the last 12 months, et cetera, et cetera; then if you reported 3.77 (but actually had 3.44), reported graduation in May (but really graduated in June), and so on, you would have scaled the minimum hurdles and despite the discrepancies, would be fine.
I'm torn on what to advise you to do and the wide dispersion in the previous comments (not something that occurs frequently on this forum which should tell you something) should underline how murky this is:
+ On the on hand, keeping silent feels best if you believe this is merely a credentials check and not a fact-finding mission. Was a minimum GPA specified when you applied? Is it above your cumulative GPA?
+ On the other hand, though, if this is an application check (where information you have submitted, say, on an online form) where checked, any recruiter would find the sizeable discrepancy rather difficult to explain. Yes, in such a case, proactive disclosure (as someone - or a few people, actually - have mentioned), but you would really need to have an airtight explanation (I'm personally not swayed by the one you claim) AND a credible composure to make it work.
Perhaps you need to find out what checks are being expected and proceed from there. Once can only hope that the required GPA is below your final cumulative because if it is not, it could cause rather sticky situations in the future.