I failed like 2 shinies using the synchro machine in indigo disk and I just wanna know if you can make it so that the Pokemon dont despawn once you go out
There's no way to enter a conventional battle or throw Poke Balls while using the Synchro Machine. If you want to catch a shiny Pokemon you find while under its effects, you'll have to lure it back to where your character is, stop the synchronization, and engage in battle to catch it.
Synchro Machine (Bulbapedia) Synchro Machine Tips (Reddit)