For both bows the general behavior is the same: you cannot swap cancel the proc condition skill (Scatter Shot for the BRP bow and Acid Spray for the AS bow), as that will cancel the proc effect.

You can however fire Scatter Shot from any random bow, swap to a BRP bow, and as long as you are on the BRP bow bar when Scatter Shot impacts the target, the Virulent Shot proc effect is applied.

The same is true of the AS bow. You can fire Acid Spray from any random bow, swap to an AS bow, and as long as you are on the AS bow bar when Acid Spray impacts the target, the bonus damage proc is applied to your next Snipe/Scatter Shot/Poison Arrow.

This leads to a really weird combined effect where if you wanted to use both bows together, you might ideally put Acid Spray on your BRP bow bar and Magnum Shot on your AS bow bar. Which is exactly the opposite of how you'd logically set that up.

But in theory, this means you can ...

[BRP Bow Bar] (Various front bar stuff)-->Acid Spray-->swap
[AS Bow Bar] Endless Hail-->Magnum Shot-->swap

... and you will buff your Magnum Shot with the AS bow bonus and apply the BRP bow's Virulent Shot effect.

Setup (Bar 1 = BRP bow, Bar 2 = Asylum bow):


Virulent Shot applying when swapping from Bar 2 (Asylum bow) to Bar 1 (BRP bow) such that Magnum Shot impacts the target on the BRP bow:


Note the damage of Magnum Shot in the prior screenshot (5990) non-crit. Here is the Asylum bow bonus applying to Magnum Shot via Acid Spray on bar swap: