I am ignorant.
Is there a method of determining ancient atmospheric pressure from the fossil record?
Not directly, no. However, there are terrestrial (and flying) animals that are in the fossil record who demonstrate that the atmosphere must have been radically different by their anatomy. They are either too large to have lived in our current atmosphere (the large sauropods) or too large to have flown (the pteradons).
Some scientists have proposed other theories to account for the anatomical problems, such as proposing that the sauropods spent all their time wading in water to alleviate their massive weight, but those theories do not hold up. Sauropods had terrestrial feet, and they lived in areas that didn’t have enough water for them to constantly wade in. The only theory that seems to solve both issues and is tenable is that the atmospheric pressure was enough to give these creatures buoyancy.