Try looking at it this way. Say there was only a runner on 1st and the same thing happened. It would take a pretty sorry SK to pop the catcher, even if the ball flew into the CF’s hands. So no matter what happens, there’s no E because of the runner on 1st.
The only question in my mind is whether the runner on 3rd took off because of the errant throw. That’s pretty much a timing judgment, and a lot depends on how closely the SK was paying attention to the runner on 3rd in relation to the other things going on. To be honest, a lot would depend on where the SK was sitting.
If s/he was in the 1st base dugout, its pretty tough to be able to watch that runner on 3rd very closely and still pay close attention to where the ball was. If s/he was sitting in the 3rd base dugout, it would certainly be a lot easier to make the determination in this particular situation. If the SK was behind the backstop, where I always try to sit, a lot depends on where his/her attention is most focused.
Most experienced SKs will try to look at all the runners to see what they’re doing, even if its just a glance. But its really difficult, and not a good idea, to change focus when there’s a play going on like this one. So, if the SK is positive the runner on 3rd had stopped or had no intention of trying to steal home, then scores only because the throw was bad, s/he should score an error. But, if s/he wasn’t paying close attention or couldn’t be pretty sure, s/he should just let it go and score the double steal.
But for sure, the last thing the SK should take into consideration is what anyone on the field was yelling or telling each other.