We report measurements of radon progeny in liquid argon within the MicroBooNE time projection chamber (LArTPC). The presence of specific radon daughters in MicroBooNE's 85 metric tons of active liquid argon bulk is probed with newly developed charge-based low-energy reconstruction tools and analysis techniques to detect correlated Bi214-Po214 radioactive decays. Special datasets taken during periods of active radon doping enable new demonstrations of the calorimetric capabilities of single-phase neutrino LArTPCs for β and α particles with electron-equivalent energies ranging from 0.1 to 3.0 MeV. By applying Bi214-Po214 detection algorithms to data recorded over a 46-day period, no statistically significant presence of radioactive Bi214 is detected, and a limit on the activity is placed at <0.35 mBq/kg at the 95% confidence level. This bulk Bi214 radiopurity limit-the first ever reported for a liquid argon detector incorporating liquid-phase purification-is then further discussed in relation to the targeted upper limit of 1 mBq/kg on bulk Rn222 activity for the DUNE neutrino detector.