Robin2:
I suspect it would be much easier to achieve that with a low-geared DC motor that turns a crank with a throw that exactly matches the movement you need. Study the sort of arrangement that was used decades ago in windscreen wiper systems - long before electronics were used in cars.With a stepper motor you will need at least one limit switch so that when the Arduino starts it can determine the HOME or ZERO position. You also need arrangements to deal with the risk of the Arduino losing count of the position and the stepper going too far in either direction.
...R
if you use a stepper and a cam
one revolution can push the flush handle.
the one home button will indicate that a full revolution has been made.
this is pretty simple and counting steps in not needed.
if you chose a different mechanical linkage, the same applies.
one revolution and stop. the switch is both home and end and Bob's your Uncle.