Catalyzes the formation of cyclic 2,3-diphosphoglycerate (cDPG) by formation of an intramolecular phosphoanhydride bond at the expense of ATP. (2R)-2,3-bisphosphoglycerate + ATP + H(+) = ADP + cyclic (2R)-2,3-bisphosphoglycerate + phosphate Belongs to the cyclic 2,3-diphosphoglycerate synthetase family.