A complete data-sheet would ideally have a curve of air flow vs pressure difference. Likely the shape of the curve would be similar (like a more or less straight line between the extremes), but with different scales. So it is about the two points of free air flow and the peak pressure. The interesting part would be the airflow at an somewhat intermediate point.
I don't think there will be simple relation with rotational speed to the pressure. It is also the number, size and shape of the blades and the angle.
Measuring pressure difference or temperature (difference) sound like a good idea, especially if you can still run a test with the original fan. I am not sure about the pressure range, a simple version with water in a tube could be good enough.
For the DMM the important part is to get a similar airflow, as there is not only one temperature, but also temperature differences that depend on the air flow. So to get the ultimate precision the airflow should also be constant - thus the usual procedure of cleaning the filter. A temperature regulated fan is not such a good idea, as it might initially look. This would only work well if the critical parts where enclosed without direct air flow.