If holding the tuner button makes it go away, I would put money on it is something vibrating in the tuner itself. Sometimes if the button gets a bit worn, it can be slightly loose even with the screw tightened all the way down (a tiny washer might fix that problem if it worn under the screw head/ some plumbers tape perhaps might fix help if it has wear on the inner hole that goes over the shaft). If the base plate gets bent, worn, damaged, it's possible that it is loose where it comes together (taking it apart and straightening any bent parts might help this). Sometimes, there's nothing you can do but replace the tuner. If that's the case, a rubber band placed on the tuning button and the one adjacent to it may provide a temporary fix and keep it from rattling around. I used a twelve string with six strings for a long time and I had ot do this to keep the unused pegs from rattling. Also depnding on where the rattle is coming from, a bit of beeswax (can use the stuff that comes in a wine bottle) can prevent rattle. FOr example if the button is loose on the shaft, coating the inside with beeswax can stop the rattle.
Mysterious vibrations can be a nightmare to sort out. Good luck isolating the root cause.
One thing you might try: It may be that when you hold the button with your fingers, the pressure changes something and stops a rattle from somewhere else or even just adding the extra mass to the headstock you might be affecting the transfer of a vibration that has nothing to do with the tuners. Try applying very light pressure to the button with a chopstick or some other item. If it's the button itself and not something somewhere else it should stop. If not, then try pressing in various parts of the tuner to see if that makes it stop. When probing like this, if it stops when applying a little pressure directly to a particular part/piece of the assembly, then that is probably where the problem lies. (sorry if that's confusing. It's late and this is a much easier thing to show than explain). Also check the tiny screws that hold the base plate of the tuner assembly to the headstock to make sure you don't have a stripped loose screw (probably the one closet to the G-string in this case).
If all else fails, try removing the string and putting it back on.
One Ring to rule them all, One Ring to find them, One Ring to bring them all, and in the darkness bind them.