There are existing instruments commercially available to measure hemoglobin-oxygen saturation curves. Whole blood or lysed blood can be used; I would expect the instrumentation to come with specific directions for sample preparation because it may depend on the exact equipment.
These devices use a spectrophotometer to measure saturation, taking advantage of the same color difference between arterial and venous blood that is visible to the human eye.
To collect a curve, you start with the sample saturated in room air or oxygen, and the instrument gradually introduces nitrogen to replace the oxygen, while also monitoring oxygen concentrations using a Clarke electrode.
Here is an example paper that uses one of these devices. Their precise methods may or may not be appropriate to your research approach.
Head, C. A., Brugnara, C., Martinez-Ruiz, R., Kacmarek, R. M., Bridges, K. R., Kuter, D., ... & Zapol, W. M. (1997). Low concentrations of nitric oxide increase oxygen affinity of sickle erythrocytes in vitro and in vivo. Journal of Clinical Investigation, 100(5), 1193.