Its easy to work around it for small time pulses, but yeah 30mA for 100ms requires a crazy level of capacitance,
the ESR of a CR2032 is about 20 ohms at room temperature, down at 0C its close to 75 ohms,
I've built my own little RF modules, but I went out of my way to keep the RX/TX time as short as possible (5.5ms @ 11.7mA in my case) That still required a 47uF in parallel to ensure it could transmit down to 0C, your using much more current, for a much longer time, so yeah 220uF will not cut it at all,
To work it out, you treat your consumption like a constant current load, with the battery being a voltage supply with 75 Ohms in series, with X sized capacitor in parallel, then work out the size to keep the voltage above your cutout voltage (1.8V in my case), I ended up using Falstads circuit simulator for this, but there are ways to do it with math.
For me that 47uF let me discharge the battery all the way down to 2.3ish voltage and able to handle the transmission,
http://data.energizer.com/pdfs/cr2032.pdf