Best method for trasmitting multiple video feeds at once via Bluetooth or WiFi?
You could transmit video as transport stream which is a common format used in DVB.

If you want to use a microcontroller, you should know that you need a powerful one, maybe some development board with an Embedded Linux operating system. Ready made devices that do that are used by DTH and DVB-T providers to uplink data to satellite transponders or terrestrial multiplexes. But this "muxers" provide raw data at their output which must be fed to a PSK or OFDM modulator then to a transmitter.


  save on bandwidth, increase speed, and maintain quality



Speed depends on connection (link) speed for example 24 MBit/s for Bluetooth 3.0 or 54 MBit/s for WiFi. It is comparable with the bandwith of a satellite DVB-S transponder at a Symbol Rate of 22000 MS/s (30-40 MBits/s).
Bandwidth and quality depend on the codec (for example MPEG-4 is better than MPEG-2)


If you want to try with a PC, you could use VLC and connect with another device over WiFi.