The TL-4903 is an AA cell, and the nominal voltage is 3.6V, If you can use a low Iq buck instead of the LDO it might be a better solution, or better yet, power your board directly from the cell.
The cell should be able to provide the 16mA pulses without any issues. I used the same cell with 100mA pulses with 470uF input capacitance and the voltage dropped to ~3V in the worst case, -20C and an almost dead cell.
If possible, use the largest buffer capacitor you can fit, Google can't find that Murata PN, but there are 220uF in 1206 and 1210.