Remote transmits (also beeps and lights LEDs) for ~975ms at 48mA with 3.2V supply. See screenshots 1, 2, 3 below. This is 150mW power and 0.042 mWh or 0.013 mAh energy.
Because F = As / V, if you allow 0.5V voltage drop during the peak, required capacitor would be
0.013mAh * 3600 s/h / 0.5V = 93600µF.
Not going to happen.
Standard electrolytic, tantalum and high energy density MLCC capacitors are good for peaks up to some 1ms.
Now if this was 1ms instead of a full second, a 100µF 4V part would work. This would still be smaller than the battery itself.
This is why many communication devices really require low-impedance battery sources, unless the packets are really short. Power density of capacitors is great, but energy density is like 5-6 orders of magnitude behind batteries.
Still people often solder capacitors as cargo cult attempts to solve these issues, without performing simple napkin calculation first. The result is obviously disappointing. For the same reason, you may see unpopulated capacitor footprints on the PCBs.