A posturally related cough suggests that the cough may result in a shift of internal structures, to press against nerve endings that are part of the cough reflex, when you arise, to sit or stand. That would be hard to prove and detection of abnormalities within your chest might require further (X-ray) imaging, specifically a CT Scan of your chest.
By definition, any cough that persists for 3 months or longer is considered to be a chronic cough. One type of cough that routinely persists for a long time (3-6 months) is that caused by whooping cough (pertussis). I believe that it would be reasonable to ignore the postural changes you describe and just approach this as a chronic cough of unknown cause. There are established protocols for a diagnostic assessment of chronic cough and, in addition, most university medical medical centers have Cough Clinics staffed by physicians with experience and expertise in the diagnosis and treatment of chronic cough. You might want to discuss this with your doctor.
Good luck
This makes sense. I am going to share your input with my doctor and hopefully that will help with the diagnosis.
Thanks so much
knaderi