Many shops will plug the fingers and they set overnight,
Next day the thumb and it sets overnight,
Any balance holes (depending how far away from other holes) may need one more.

The question is do all the holes need plugging?  Sometimes the finger holes can remain where they are and just change inserts (assuming pitches are close and layout is desirable).  If you keep the same layout you may not need to plug the balance hole either.

And yes, you can plug a ball with the IT installed.  That particular hole is large enough that it probably should be plugged 1/2 at a time to avoid the plug material getting too hot and cracking.

Could Take up to 3 days depending on what I'm using and how fast I get it. I also use Vise CS Hybrid so I can fully plug a ball in an Hour and have it ready to drill. I only recommend quick plug if its a last min or a screw up. If you have the time, go a head and let your PSO keep the ball for the time required.

The IT can be plugged easily plugged in one stage. Drill out the molly and prep the edge of the hole. I've done it numerous times with no cracking all the way up to a 1.5" Switch Grip Sleeve. I don't advice drilling directly back into those holes unless you've let it set up for a couple of days.