Test it to be sure and let us know. Here's a simple way to test:
1. Fire destro ult on your destro bar and wait until it starts ticking, then swap to SnB bar and record the noncrit damage.
2. Fire destro ult on your destro bar, but this time swap cancel it into SnB the way you would a DoT to have the SnB bar be the "impact bar". Record the noncrit damage.
This should tell you whether the destro ult works like a single target dot or like a ground target AOE dot. Below is why I'm suggesting that you test it this way. The context of this chart is for stamina characters, but it describes the mechanics of single target and GTAE dots, which can be pretty weird. My theory is that you will want to swap cancel the destro ult for best effect.
What that means:
Second bar trait IS IMPORTANT as Penetration carries over from the bar you are on when the dots hit the target. So your dots will do more damage from a sharpened bow compared to another trait even on the dual wield bar.
Swap Canceling Poison Injection means the “Impact Bar” of Poison Injection is Dual Wield, which increases your dot damage for the whole duration.
If you cast a Beast Trap on your Bow bar but it activates on the DW bar, DW is the Impact bar.
If you cast a Poison Injection while Warhorn ran out and Warhorn gets applied afterwards, dot damage of PI will not increase.
Let's say, I am running -
5PC Body Clever Alchemist
2PC Body Valkyn
3PC Jewelry Sun Silks
2PC SwB Sun Silks
1PC Destro Sun Silks
When I pop the destro ulti, then switch bars, does the 5 PC Silks of the Sun affect the destro ulti?