First of all - this is not a case of the wood - it is case of the construction technique. Totally different.
Also - Wood DOES matter, but just a little - just for your "bedroom sound" - Not audible in a gig. AND - any wood differences are negated by a touted "High Mass Bridge" that prevents the wood from vibrating. If you have a resonant piece of wood, you want the lightest possible bridge. High Mass Bridges are used now because there is less and less "Good Wood" and they help the instruments to have a more consistent sound quality as the wood varies.
Which means if you want to upgrade a cheap bass, those usually come from lesser woods.
Get a nice phat chunky bridge.
Good tip. I'll remember that.