I would advise to get a instrumentation amp like AD623 or AD620 or equivalent from your favourite supplier, attach a gain setting resistor that matches your 1 mV/mmHg requirement (the rule 5µV/mmHg per volt excitation voltage is right) and, uhm, prepare offset injection as there is both a manufacture-related offset (0mmHg output different from 0µV) and also some temperature coefficient, and you also see a different zero signal depending on the column of water above your sensor. Patient monitor manufacturers in FDA land are required to give or take 150 mmHg to compensate for offset iirc.
Please use a rather stable excitation voltage source capable of some mA, and remember your signal is proportional to your excitation voltage, that means, ratiometric. You should at least be able to tell what excitation voltage is there if you want to tell the actual pressure.