Re: "How to use a barometer to measure the height of a building."
During the 1960s the History of Science mob at Cambridge, IIRC, actually tried Galileo's thought experiment though with a local tall building, not the Tower at Pisa.
They were unable to determine exactly when the balls hit the ground because they were moving too fast.
Thus use a barometer, use a girder, probably doesn't matter.
Use a bit of girder on the end of a string as a pendulum would probably be quite accurate enough.
The barometer on my phone is sufficiently good for me to distinguish which way I went into town (one way being all downhill but the other has a bump) so using a mercury barometer should be quite accurate provided you have a calibration chart.