Really, it has more to do with your "attack" then it does your gear. It might say Frank was using an ESP - but God only knows the modifications that were put in to it. Artist will receive instruments from dealers for free, may even get paid a little to endorse the product, may have a signature series... but the one they play on stage every night is not the one that you are going to buy from the store. They are free to change the pickups and whatever else they want as long as that headstock still says "Fender" or "Jackson."
Anyway, back to the question! Frank has a wicked attack. He almost always plays with his fingers, but he plays HARD. He slams on the strings. That is step one to getting that clackity sound you hear on ATL. Having a Bass with a Precission pickup helps. Play over top the pickups with your fingers. Having an amp with an EQ helps a little, as you can adjust your mids just so to achieve that style of sound. Not much los or his but lots of mids.
So, turn up the precision, drop the pick, and beat those strings like a red headed step kid with lots of mids!
p.s. Depending on how desperate you are, Frank's signature bass achieves this sound right out of the box. The pickups on that thing are LOUD and very HOT. And it is a very solid instrument for the money.