I understand how and why low pressure systems are often associated with rain, but does the rain itself (both during and after) alter barometric pressure in any meaningful way?
Under a hydrostatic assumption the basic equation is
p=/rho R T. So under this easy equation the pressure depends to the air density and the air temperature. As you may know diabatic processes during rain yield to a decreasing temperature. but the magnitude in change of pressure against the synoptic scale forcing is low.
Please be aware of mixing stuff together answering this questions. The question was not to answer how precipitation is forming. It was just the question about the rain influencing the pressure.
No. Low pressure affects the rain, not the other way around. Air at high pressure/density can hold more moisture than air at low pressure, so when there is a reduction in pressure the moisture content, or some of it, condenses and is precipitated. Reduction of temperature can also cause precipitation, because cold air holds less moisture than warm air. Reduction of temperature can be brought about by lowering of pressure.