The impact of SS on the recast of Burning Strike seems to be very close, if not the same, as on the player. At 737 SS, I was getting 2.83s recast on Burning Strike; at the same SS, Shadow Flare is a 2.82s cast. That said, it seems like most of the impact is lost in practice.

There's an arbitrary delay between each Burning Strike, and it was consistently shorter with 354 SS than it was with 737 SS. When the it was on the shorter side, it nearly evened them out; the 737 SS set only pulled ahead when the 354 SS set had a relatively long delay.

https://www.youtube.com/watch?v=Zj0mcBs97eY

If both sets had an equal delay between Burning Strikes, the higher SS set would win, but as it is, it seems like it only pulls ahead occasionally. It's as if the pet's AI doesn't know that it can attack more frequently than 3s. As such, the only consistent benefit from SS seems to be the impact on the longer recast attacks.