how powerful a stepper motor do i need
Note that the amount of force (Newtons) available from a motor depends on how far away from the center of rotation you are, because torque is the product of force and distance. (We measure torque in Newton-meters or pound-feet or somesuch unit.)

A gearbox or pulley will give you close to the exact ratio of torque increase; a pinion with 12 teeth driving a sprocket with 60 teeth will give you about 5:1 torque increase, with the corresponding 1:5 decrease in rotational speed. Same thing for a belt or any other kind of rigid ratio.

Regarding "not skipping steps," you will want to add limit switches no matter what. Even with brushless motors, as suggested in the comments, you need to know how far to move, and unless the BLDC has an absolute encoder attached, you can't guarantee the exact stop position without limit switches.

Stepper motors are reasonably tolerant to light over-volting, or even over-current-ing, because they have very few moving parts, and no brushes to wear out. Also, if you only run them for a short amount of time, there won't be as much heat build-up, and thus you can run them at higher current for a shorter duty cycle. If you can crank up the voltage, and turn the motor driver up to maximum current, that might be enough. Or not -- the L293D is not a high-performance driver.