Powering microcontroller with a capacitor
Sounds plausible.
Maybe you could use a Qi transmitter/receiver to produce 5V and use a simple LDO regulator down to 3.3V. I'm not sure they would switch on fast enough and give enough current for long enough to work. You would need (say) 1A for less than 500msec to keep an average 40mA going, which sounds to me like it might be worth investigation.
Regardless of how you get 5V to the target, a super-capacitor such as the AVXSCMT32F755SRBA0 7.5F 5.5V with 90m\$\Omega\$ ESR might do the trick (charged to 5V and with an LDO regulator) with room to spare. If you use brushes, maybe you can inhibit communication during the noisy bits and add a current limiter to the 5V supply to keep the peak current tame.