So I'll try a trick... I cannot test it, cause I've disabled burst mode in some Xposed module and didn't find the option to enable it again, so please report if this thing works:
I think, longpress at touchscreen at camera button should also result into a burst shooting, so here we go.
This task will emulate a minimal swipe at the camera button --> long press.
So, default long press is set to 5000ms = 5secs...
My SE MW600 bluetooth just won't work on long press(next/play/previous button) with tasker, either single shoot or burst shooting, all no.
Can you test the long press on your Sony SH20? If it works with long press, then I will buy one.