This is a complicated question...but I'll try to help! One thing that you have to realize is that it's very difficult to completely replicate any artist's tone without the exact setup they are using, as well as their playing style. Everything from your hands to the speakers has an input on your tone. These guys are pros, and they are playing through world-class gear from beginning to end. AND they are on TV, which has a LOT of post-production qualities to it. Meaning that it's not just the amp and bass, but also a lot of signal processing after the engineer recieves the signal and sends it out to broadcast. And in broadcast, there is a lot of compression, limiting, EQ and other "stuff" that you end up hearing that may not be present "in the room"... In addition, when you are hearing a bass tone in context of an ensemble (read: band), the way the bass is communicated to your ears is different than when you are practicing or listening to just the bass track played back on disk. You have to analyze your tone by itself, but also in relationship to the other instruments that are playing with you.
That being said, this guy's tone is pretty much stright up P-Bass + Ampeg style tone. There are many ways to approximate that tone on tape. The BDDI is not really an Ampeg tone. It's a BDDI tone! Which we usually credit Geddy Lee of Rush for making popular... But the BDDI is and has been a standard backline DI for bass players for decades, so you will feel that the tone that the BDDI generates has a familiar feel to it.
To more closely approximate his tone, I might recommend that you check out some other pedals like the Tech21 VT Bass and VT Bass deluxe pedals... Here are some links: Tech 21 Character Series VT Bass Pedal at The Perfect Bass Tech 21 Character Series VT Deluxe Bass Pedal at The Perfect Bass
We ahve found these to be very popular for bass players looking to get that classic SVT tone and we agree, it's probably one of the best ways to replicate it short of buying a Heritage Series head and cab! Ampeg Heritage Series SVT-CL Amp Head
That's a link to the Heritage Series (USA Made) SVT-CL head. We have a video there with Dino M from Ampeg talking a little about it as well.
Plug ins can help also, but I have found that while they are generally really good at what they do, nothing beats having as close to the tone that you are seeking going into the board than actually having it! So I recommend trying to get your tone as close as possible to what you are looking for, before you mic up or plug into the DI on your recording rig.
As for the pick question. Yeah, it's hard to replicate that pick style attack...without a pick...
There's a place for using a pick and a place for not...just depending on the tone you are looking for. Generally though, that sharp attack on the front end of the note that he has in the video, is pretty much attributable to him using a pick. You can get close with various right hand techniques, but a pick tone is a pick tone, and if that's what you want, you should prectice with and without a pick. I would not recommend just going one way or the other, but that's a personal choice. For maximum versatility as a player, I would think you would want to master both techniques.
One of the beautiful things about making music is that it never ends... We are all constantly looking for new sounds, and there's an infinite number of them out there to chase... HAHA!! It's a lot of fun... I hope this helps and rock on!