Dealing with echoes / dead-spots in digital transmissions
If the channel is static and you are in a deep fade -- then, yes, your only option is to increase the bandwidth or go multicarrier like you suggest.

In fact, modern WiFi systems use that exact technique: OFDM ("Orthogonal frequency division multiplexing" -- sometimes known as multi-tone or multicarrier transmission).

Alternatively, you could go with two receive antennas (microphones in your case) spaced far enough. If you use 2000Hz carrier, the wavelength is v/f = 300[m/s] / 2000 Hz = 0.15 m. So a distance of around 10cm should give you enough diversity. This can also be implemented at the transmitter.