mouse - AHK script to click a specific button in a program not working inside the program window?

I'm trying to create a hotkey for a software button inside a program since the devs of the program did not do so.

Here is what I tried to move the mouse, click the button and move the mouse back to the original position:

F3::
CoordMode, ToolTip, Screen
MouseGetPos, X, Y
Click 512, 516
MouseMove, %X%, %Y%
Return

This works, on say the desktop, but when the program window is active, nothing happens. Is there some way to make this work inside the program window?