My firm used to do this. Complete reimbursement of your bills, up to $110/mo, but you had to submit every page of the bill. I switched to a cheaper plan which does not provide an itemized bill (Metro PCS). I was expensing way less that everyone else. Boss just thought I was cheap. He didn't realize that what I was actually doing was protecting my privacy. And I'm frugal, not cheap.
Eventually, they issued everyone Blackberries and tried to make us port our existing phone numbers. (Really. In about 2010-11. Funny, I know.) They wanted to be able to keep your contacts in case you left the company. FU, sez I. I kept my old phone and number. Left the BB in the car all day, in case the company called or messaged. Even my boss never called me on the new number. From then on, I paid my own phone bill and wrote it off on my taxes. When I hit FIRE at the end of 2012, I left with my phone number, contacts, and privacy intact. Totally worth it.
Back to you: the fact that the books are "closed" is bullshit. This error was not your fault. They should find another way to compensate you. If not actual dollars, how about some PTO?