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.
My opinions are the result of years of rational, objective analysis. I analyze all factors before making a choice. I update my opinions to include new facts. Fallacies? No?