Determining the required threshold for a filter
I think you would do better to fix the noise at its source rather than trying to remove its effects from the output.
You are using an Arduino, so you've got 5V available somewhere.  Take that 5V and put it through a low drop out 3.3V linear regulator.
Your module will work at that voltage (the documents say it will work down to 2.7V.)
The regulator will cleanup the noise going into the module, and you won't have to try to clean up the output of the module.

As to measuring the frequency of the interference, you can use the sound card of a PC as a makeshift oscilloscope and (audio) spectrum analyzer.  There's free software available, and schematics available to make adapters that will (to some extent) protect your computer.
I use a 50 year old analog oscilloscope for most things, and sometimes my PC as an audio spectrum analyser.