I am using a 750mAh 25C 3.7v LiPo battery for a custom electronics project which currently draws ~1.5 amps, far below the rated operating current of (0.75 * 25 = ) 18.75 amps. However, at the battery terminals (before any PCB traces) I am noticing a drop from 4.03v to ~2v, below the 3.3v I require for my MCU. Why is this drop so significant? Is the battery bad?
Assuming it was fully charged before, either your battery is bad or its rating is hugely overstated.
LiPo batteries should never be discharged below 3V, but for longevity it is best to have cut-off at 3.2-3.5V
Note that 1.5A is relatively high load for only 750 mAh battery of any C rating. I am not saying they shouldn't be used this way (RC toys and electronic cigarettes use same batteries at currents many times more than that) but cell lifetime will be affected.
The usual definition of C rating is "how fast the battery can be discharged safely". The safety bit should be understood as not blowing up or exhibiting immediate drop in performance. This does not mean, however, that discharging at full C rating has no negative effects. In fact, by pushing the cell to its limit you severely reduce its lifelong capacity. I've seen many 400 and 750 mAh cells used in RC helicopters completely giving up after only 50-70 charge-discharge cycles.