Is the rattle just when you lift your finger off or whenever you play the open string?
The first could be technique, in which case it will improve as your technique develops, especially damping.
If the second the nut could be cut low, still not enough neck relief, action too low especially if your playing style is quite aggressive) or a high fret. Hard to diagnose without the bass in my hands!
You said you checked for high frets with a credit card. Something longer such as a 150 - 300mm rule is better.
In the short term raising the action a little if you find the rattle intrusive.