Could this have anything to do with your having worked as a temp-to-hire part of the time you were there? I am rather mystified about what could have you and your former employer so at odds on these dates, why they refuse to correct them, and how this could have cost you "a career" as you put it.
I agree with the prospective employer, if I checked hire dates and got something this variant, it would definitely cause me to rethink the job offer, but then what it gets down to is that what you showed them as verification of employment dates, they didn't think were valid enough. Why was that?
It sounds as though a very cursory effort was made by you to correct this information, and now you think you're going to what??? Report your former employer to the Bureau of Fairness and get them fined or something? Under the Fair Credit Reporting Act? I don't quite follow that at all. While you could theorectically sue them I think this would take a lot of time effort and money you may not have at present, since you're not working. Your state's Department of Labor may be able to help you with information about this issue.
Have you filed for unemployment insurance after working for the last company? If so, your benefit eligibility statement would demonstrate whether you had earnings from this company in the previous 18 months. Your company couldn't very well deny those dates if they were shown on your W-2 and your paycheck stubs (if you have those). Are you positive there's not a staffing agency involved here?