It looks like a Selita SW200 movement. It's essentially an ETA 2824. If the keyless works aren't botched, you should just be able to reinsert it, turning slowly while pressing in the stem in a tightening (clockwise) direction. With an A7750 movement, you have to press the stem release button while you reinsert the stem, but you shouldn't have to on the SW 200.
From HERE:
"...quite the contrary. imho the 2824 has possibly the pickiest keyless works i have ever dealt with. anytime a movement is giving me an issue with stem re-insertion I usually try this: Try pressing down on the stem release button when re-inserting it all while giving the stem an every so gentle twist. This usually helps you realign the stem with the sliding pinion. hope that helps"
...this is not likely the proper way, as from what I understand, the Sellita does not have the stem release button pressed on re-insertion.
Any watchmaker should be able to reinsert it for you...it's a Swiss movement...so no worry about them needing to be rep-friendly...just tell them you have a Sellita SW 200 that needs a stem reinsertion if you have to do it.
You can go to SELLITA for technical info on the movement.
Good luck.