The ombudsman made its decision following advice from from a consultant gastroenterologist and a consultant radiologist.
It found a diagnosis of mesenteric ischaemia should have been considered as a strong possibility based on the symptoms.
A CT scan also failed to report the narrowing of the blood vessels supplying the gut, and therefore it found the decision to discharge C was unreasonable.
The health board was told to apologise to C and also reimburse them in relation to private treatment, including the cost of travel to and from London.
The calculation will be based on what the treatment / surgery would have cost the NHS, rather than the full cost of the treatment, and what proportion of that C had to pay.
It recommended that care be taken when discharging patients with ongoing and persistent symptoms, that mesenteric ischaemia should be considered in cases of weight loss with no apparent symptom, and that here should be interdisciplinary working between teams to reduce the risk of a missed diagnosis.
A spokeswoman for NHS Ayrshire & Arran said: “We are sorry that the care experienced by patient C did not reflect the high standards of care we aim to deliver in NHS Ayrshire & Arran.
“The board accepts in full the findings and recommendations of the Scottish Public Services Ombudsman (SPSO) and will be offering a formal apology to the patient.”
They added that work was underway to implement each recommended action.