I don't think this is 100% necessary. Yes, I see the part of having a cool model to flex on people and to reward your SEALs for being good, but you already change your name and jobs to DEVGRU.
(Might have changed from when I played, or I just didn't pay attention to the Seals but) most of the time it seems like the other Seal Teams and ST6 stay together/ go to the same objective so acting like your own group seems pointless.
When it comes down to it: ST6 are still seals; they all wear the same trident, and to me, it really doesn't seem necessary to alienate ST6 from the rest of Seals.