You may get a nasty surprise trying to use really wide Forstner bits in a pillar drill. Some thoughts:
1. The thing that really matters (assuming it's a good quality Forstner in the first place), is the speed of the rim. Obviously for a big Forstner, you need a SLOW speed on the drill. That means bigger pulleys, and more torque at the business end (because of the gearing). If it has a Morse taper for mounting the chuck in the quill, that could be problematic.
2. Any run-out on the drill will be worse at the rim of the Forstner bit, causing vibration. See (1) for the problems that causes.
3. A Forstner is basically a one-tooth circular saw blade. It will blunt fast, and as it blunts it will overheat, which will ruin the steel, making the problem doubly bad.
4. A (good quality) hole saw will not blunt as fast, but depth is usually limited. It will not be as badly affected by run-out, as there are no horizontally-cutting teeth (if you see what I mean). Holesaws are a LOT cheaper than Forstners for the same quality. Holesaws still need a slow speed, because of the rim velocity. Cutting vertically with a holesaw, it is unwise to use higher speeds, because you'd need a higher feed speed to cut nicely, but you can't clear the sawdust easily (every one I've seen has no provision for sawdust clearance, only continuous teeth).
So I'd try a holesaw, especially as the hole will be hidden by the clock face. Cheaper, easier, but not without its own problems. Use a vacuum to clean the cut regularly and don't try to do the whole thickness in one go (unless it's thin, obviously). In a pillar drill you shouldn't need the pilot drill incidentally.
Plan (b) would be a router and a circle template.
Plan (c) would be an expansive bit in a long-throw hand brace. They need practice, but you have some control, and you might make life easier by perforating the hole area on a pillar drill first, especially the quadrants which go from along to across the grain (leave some in the middle for the leading screw of the bit!) - you don't have to use power tools!
Plan (d) would be the Forstner bit in a pillar drill. I have a set and a number of singles, I know how to sharpen them well (not easy in itself!), and it would be my last resort!
E.
PS: You can get extra depth with the holesaw by stopping to split out the waste when you reach its maximum depth. For that I probably would use a pilot hole (doesn't need to be central, only close), and drive in a 1/4" chisel through the side of the hole, across the grain, to help it split. You need to arrange the grain running towards the pillar of the drill (or close to that), as you can't take it off the table between "bites" (to keep concentricity).
