Catalyzes the formation of cyclic 2,3-diphosphoglycerate (cDPG) by formation of an intramolecular phosphoanhydride bond at the expense of ATP. It is also able to catalyze the hydrolysis of cDPG but with significant slower rates (8-10 times). May be involved in thermoadaptation. (2R)-2,3-bisphosphoglycerate + ATP + H(+) = ADP + cyclic (2R)-2,3-bisphosphoglycerate + phosphate Optimum pH is 5.0-7.5. Optimum temperature is 75-80 degrees Celsius. Homodimer. Belongs to the cyclic 2,3-diphosphoglycerate synthetase family.