Small low power circuit power domains inrush
Small coins cells may have an ESR of 1~3kΩ , but I can only guess what you have.
You can use ceramic caps which although not as good as FILM but low enough leakage  often C＞0.047µF:More than 500Ω·F   so 1uF = 500MΩ....in good quality ceramic (eg Murata)
use a 1:10 cap divider ratio to reduce load to source regulation error.
e.g. 1 uF source , 0.1uf switched load and  add ferrite beads to reduce Voltage dip further.