Will fitting a oil cooler drop the oil pressure.

shouldn't do (the same was a small empty bottle has the same air pressure as a big empty bottle)

If anything it will maintain oil pressure due to the oil not getting as hot,

I don't think it is the same as the airbottle. The oil has to be pushed through the pipes and the oil cooler, so this will cause a pressure drop.

An oil cooler is a ressistance to flow ie it causes a pressure drop accross the cooler and slightly reduces flow rate. But because it reduces oil temperature and therefore increases oil viscoscity which will increase the gauge pressure.

Oil pressure isn't critical --- but if it changes it is a symptom
Oil flow rate & temperature are critical.
Cold oil kills more engines than hot oil, never fit an oil cooler on a road car without an oil thermostat..