Are you checking fret buzz acoustically or amplified? You should only be doing it when plugged in so you do not have to pluck ay harder than you would when normally playing. You also mentioned it does it when you are doing plucking more than one string at a time, how is your technique? You could very well be moving your hand when you add the second finger on the fretboard causing the fretboard.
If you are still getting buzz, give the bass a setup. It can definitely be done yourself with the guides found on this site, but if you feel like wasting money, go pay for somebody else to do it.
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