I'm not sure a Forstner bit is the best way to go here. You'd want one the same size as the existing hole so it didn't wander around (with no wood in the middle for the central point to dig into) unless you used a drill press, and the small spurs on the outside of the drill would make a thinner, weaker area around the circumference of the hole.

I'd simply get new pots with longer shafts. A pot upgrade isn't a bad thing, so you might as well do it now. Guitar pots normally come in three threaded section shaft lengths, short (6mm), medium (9.5mm) and tall (18mm) (values given for CTS pots but other makes are almost identical). Short are meant for scratchplate mounting, medium for most rear cavity mountings and longfor rear cavity mounting with a very thick top.

I'd measure the top depth to see whether a medium or a tall shaft would be the best fit. Tall will be excessively tall, so you'll need to adjust the amount of thread poking out of the top with a nut and washer on the shaft on the inside (so make sure any tall shaft pots come with two nuts and two washers).

Knobs come in either 18 or 24 spline versions (I have found sone Strat-type knobs that are a universal fit and they are very good) so you need to get knobs that fit the pot shafts. CTS and Bourns pots are normally 24-spline and Alpha pots 18-spline (kit pots will be 18-spline), but they all make 'the other type', it's just finding stockists with supplies.