I did a few (three) take out's and put back's just for fun on a scrap ETA2842 (from an automatic swatch). My big mistake was that when trying to rotate the regulating pins, I did it the other way and obviously I bent the spring :) But then I got to straighten it and make it work (and now I get to tell the story) - I mean I made it work, not made it work perfectly ;)
Anyway, my way of putting back the stud into that "fork" was: put it into position, with your left hand hold the movement holder (with the movement in it) and with the right hand use the back of a pair of tweezers (or other big, inelastic weight) to push it into the fork. Don't do it with the tip of the tweezers because it's somehow elastic and if it slips, you will break it again. You need controlled force. So rest you hand on the table and control your movement. I did it this way.
I hope you understand what I'm trying to say and also I hope it works for you too.
My advice, try to fix what you broke, from my point of view it's a treat. You work marvels when you have nothing to lose. Use superglue or shellac, if you have, to put the stud back if it fell out the first time. Try to straighten the spring, watch Mark's wonderful videos on that. Sweat a little on that hairspring. At the end it will be very rewarding, even if it doesn't work, next time you will get closer. If this is a hobby, this is the only way to learn.