Short version: forces applied to solid objects move at the speed of sound in that object.

Lets say your stick is made of steel. The speed of sound in steel is about 19,000 feet/second. Assuming you could push hard enough for the force to be felt on the other end, it'd take over 18 hours for your partner on Earth to feel your push from the moon.

because...

https://www.youtube.com/shorts/k5s1cMNTmGs

https://youtu.be/6g2bHqV01es

EDIT: It's in Polish, but it's still a good video.

The speed of 'push' is effectivly the speed of sound in a medium. So your shove would be the same as propagating a soundwave through whatever that rod is made of.

Veritassium covers this https://www.youtube.com/watch?v=EPsG8td7C5k&t=61s

The whole poll does not move as end entire unit instantaneously. You send a sort of shock-wave through the poll, when you push it from your end. That shockwave has a travel time that's much slower than light. I suspect that the speed of that shockwave probably proportional to the speed of sound in the material that the poll is made of.

Long winded video about it:

'Are solid objects really “solid”?' (go-to 7:30)

• https://www.youtube.com/watch?v=DqhXsEgLMJ0
• https://inv.nadeko.net/watch?v=DqhXsEgLMJ0

There's a bunch of these thought experiments that try to posit scenarios where C is violated.

Here's one I remember from uni involving scissors. Similar to what OP was thinking, but really really big scissors.

At this scale, the stick isn't as solid as your intuition would lead you to believe. Instead, you have to start thinking about the force at the atomic scale. The atoms in your hand have an outer shell of electrons which you use to impart a force to the electrons in the outer atoms of the stick on your end. That force needs to be transferred atom to atom inside the stick, much like a Newton's Cradle. Importantly, this transfer is not instantaneous, each "bump" takes time to propagate down the stick and will do so slower than the speed of light in a vacuum. It's basically a shockwave traveling down the length of the stick. The end result is that the light will get to the person on the other end before the sequence of sub-atomic bumps has the chance to get there.

This doesn't account for blinking.

If your friend blinks, they won't see the light, and thus would be unable to verify whether the method works or not.

But how does he know when to open his eyes? He can't keep them open forever. Say you flash the light once, and that's his signal to keep his eyes open. Okay, but how long do you wait before starting the experiment? If you do it immediately, he may not have enough time to react. If you wait too long, his eyes will dry out and he'll blink.

This is just not going to work. There are too many dependent variables.