I wouldn't plug and drill, because I can guarantee when you press the anchor back in you will crack the cavity, which is why it is not and never will be an option in the list.
If the gap is over 1mm I always recommend the wood dough over CA. It forms to the cavity to fill very nicely and is 10 times harder than the wood is.
There is space for the stud to wiggle a little, but there are no visible "margins" between the stud and the wood itself. Let me try to explain as clear as possible what's the issue.
What caused this to happened in the first place was that when I wanted to set up the trem one day, the anchor was oxidized inside the stud, and when it finally came loose, the stud also gave a little way. I was as careful as possible, but there was no other way than putting enough torque on the anchor until it would loosen up. I guess that the grooves of the stud itself got some leeway in the wood of the body itself due the necessary force applied. It's also when the detuning problems began. It was something that happened gradually (the force of my heavy custom gauche working on the now loose stud in the wood...
I have disassembled the guitar now so I could take a good look. So what is basically going on, is that the stud itself in it's cavity has some leeway, but it is not observable. I can see the stud sit in the wood and besides the missing paint chip, there are no visible signs of leeway, as in margins or whatever. Ergo: it is the stud itself that gave a little way in it's own cavity in the wood.
I think that seen these facts, the way to go is definitely pulling the whole stud. Once it is pulled, I'd use the wood paste as suggested by Rich, let it dry out for some days and than my trem should be stable again.
What do the pros think?
Thanks for all the help and advice. You guys are awesome!