Google was useless, as it was giving me flights through Dubai that cost thousands, etc.
However, looking at the airport website was an excellent call, thanks. It seems that both Mahan Air and Iraqi Airways have direct flights.
However, getting the right dates is still difficult. If I fly Iraqi Airways, then I either have to spend four days in Sulaymaniyah or I have to overnight in Baghdad. To be honest, going to Sulaymaniyah is a step more adventurous, in terms of travel, than I have ever attempted before. I'm really not sure about going to Baghdad.
On Mahan Air's website, after trusting the sharepoint plugin, I can search their timetable. I've found flights that mean that I can keep to my original schedule. However, the one way flight is about two and a half times the cost of Iraqi Airways. So, I can spend four days there, or spend more money on the flight .... hmm.... Does anyone know how much there is to do in Sulaymaniyah?