I contacted the HR department shortly after starting this thread, but have not heard back. At this point I don't think I'm going to hear back. I don't think they messed anything up with the income, just the amount they listed in box 12. I calculated out what I think should be in boxes 1, 3, and 5 and my value is within 6 cents of their value, which seems close enough.
It seems that since they are under reporting it, that I will have no tax consequences if I just used the value they provide. The only thing it seems to factor into is how much more I can contribute to my HSA on my own. I am planning to square up based on my value of $5833.40 and put in $1066.60 on my own to bring me up to the $6,900 limit.
Should I list the $5,516.72 that is on my W2, or the $5833.40 that I calculate? I don't want to trigger an audit, or draw any attention to it. I won't be using the mistake to try and sneak more money into my HSA though so it seems like there is no problem with using their figure, other than I believe it is technically wrong.