Maybe your idea of where "upside down is" isn't quite right. Most likely you time your arm pull/shoulder opening one of two ways. One way is to fell the speed of your shoulders ad they fall back from the cast. You're upside down if you can feel your shoulders increasing their speed and know they've reached their fastest speed.... That's when you're upside down, but in a kinda sorta way if you define upside down the way I do.
The way I see it, upside down is the position and placement where you'd balance under the bar... or hang motionless if that's easier to think about..... while in the shape you get into for the drop. If you try hanging upside down you'll find that your head is hanging on the drop side of the bar while your feet are suspended on the pull side of the bar...... neither your head or feet seem to line up with a vertical line drawn through the bar, but your shape's center of gravity is lined up with that line.
Really, if your head and shoulders are directly below the bar, you're already about 1/8th of the way up. If you decide to open your shoulders at that moment it's going to take a brief instant to really have your opening effort take effect, and in that instant your body keeps moving for another 1/8th of the way up. So that means you're 1/4 of the way up and around with your feet pointing around and over the bar. There's no way you can make a handstand from there unless you stop pulling just before getting all the way up do some crazy arch-planche action to pause while you press back up to a handstand.
So if you feel your shoulders to know when your upside down, you're gonna have to get used to knowing when they'll be there and start your pull just before you hit upside down... Provided you have a good drop, you can do a sharp energetic pull for just a brief instant and then just coast (sorta) and open the rest of the way as your body rises under the power of that sharp pull. That's going to be easier than having to support your weight and and open your shoulders at the same time.
Another way to get the earlier pull is to do sets of three directly connected clear hips, and pull earlier on the second one than you did on the first, and more earlier on the third one.
Talk with your about this idea you came across and try it that way with spot a few times to see how early you can force yourself to go while holding tight in your shape, and see what happens. Then talk it over with your coach and go from there.
There's another way using visual cues, but try the above first since you probably squeeze your eyes shut somewhere during the skill.