Catalyzes the formation of cyclic 2,3-diphosphoglycerate (cDPG) by formation of an intramolecular phosphoanhydride bond at the expense of ATP. Not able to catalyze cDPG hydrolysis. May be involved in osmotic balance. (2R)-2,3-bisphosphoglycerate + ATP + H(+) = ADP + cyclic (2R)-2,3-bisphosphoglycerate + phosphate Activity decreases in response to phosphate limitation. Optimum pH is 6.0. Optimum temperature is 65 degrees Celsius. Constitutively expressed. Belongs to the cyclic 2,3-diphosphoglycerate synthetase family.