Usually Companies that don't provide any Salary details in their Offer letters give Appointment Letters with complete salary details & breakup after joining. Doesn't your husband have that Doc?
And how long did he work in this Company? Did he get any Salary Hike or Promotion? If Yes, those Docs would be of use to prove his exp here.
What's the size of the present Company?
Frankly, you can't blame the HR of XXX.....after all they are doing their job. In these days of everything turning out to be fake.....including the Degrees....they surely would like to take all precautions.
Another way--ask your husband to take ANOTHER Letter from his Director stating the change of address....mentioning the old one & the new one......including the dates of the locations from/to. That should help.
All the Best.