Is it a smooth oscillation or is it that the bluetooth module is drawing current in pulses? Lots of digital radios have very pulse-y current demands … you might just need a stiffer/faster regulator.
This can explain +/-50mV jumps in the signal due to changing load currents (if load changes from 0.1 … 250mA)
So I don’t think that load regulation is the problem.
Three more possible causes to check:
do you have enough dropout voltage? The input voltage must be at least 0.2V above the output voltage.
is the chip getting hot? If so, it may go into shutdown 2 or 3 times per second once it has to deliver power. This is a very small package so good heatsinking is needed.
maybe the output cap is too large for stability? remove caps, try again.