Why do you need wireless for recording?
Your wireless probably has a gate. It shuts off the signal when you're not playing. When you play, it turns on and you hear the bass and buzz.
Goingpostal is right - you probably has a shielding problem, or have interference your bass is picking up. Face 90 degrees from where you are and see if it changes.
“Please don’t make the mistake of thinking that the arts and sciences are at odds with one another. That is a recent, stupid and damaging idea..." Tim Minchin