The UK FCO states:
We advise against all but essential travel to the whole of Iraq, except to the Kurdistan Region, where there are no restrictions. There is a high threat of terrorism. This includes violence and kidnapping targeting foreign nationals, including individuals of non-Western appearance.
There are no restrictions in place against travel to the Kurdistan Region (the provinces of Dohuk, Erbil and Sulaimaniyah), as the risk of terrorism in these provinces is markedly lower than elsewhere in Iraq.
I would keep trying, plenty of operators have been in to these areas.
How many passengers and would you accept a fuel stop etc?
FBP