Old thread...I know...anyway I use an MJ-838 which has a cree MJ-259D LED driver inside. This is the same as in the MJ-808. This unit emits RF which interferes with my handheld HAM radio in the 2m band (146MHz). It is not the battery pack. I have not checked a spectrum analyzer to see what the frequency content of the interference is like. Surely it is pretty broadbanded. I have tried wrapping the power cord around a few ferrite suppressors with very little change in the level of interference. I did open up the unit to see if I could add a capacitor across the power leads to at least make the noise common mode only. There is a chip cap across the leads already on the circuit board. Maybe a series choke or something could help, but frankly I did not want to start hacking away to hard.
Will this interfere with your wireless "stuff" on the bike? Maybe. It really depends on the robustness of the data transfer protocol, the signal level, placement of the noise source, etc. The RF radiation can be coming from the lamp unit as well as the power cord.
I have found that my Cateye heart rate monitor is wicked sensitive to RF interference. It will go wonky on a whim showing 235 bpm or 0 especially when the battery is not 100% fresh.