Having done a few measurements and tried a few things, I think I can just about to work around the problems of length and breadth posed by the famous auger bit extension. You see the "chuck" is very, very ingenius, but it's also larger in diameter than my required 5/8" holes - so obviously it can't go through those holes. That also means the combined length of the "chuck", which is quite long, and the bit I use can't exceed the distance between the arm and the seat of the chair. That's about 8 1/2", which isn't much. So all auger bits are automatically out, even dowel bits (if I had one). So I turned to the centre bits; an improved centre bit with a lead screw proved to be the only one capable of making any kind of dent in the beech, but the nearest I have is 1/2". Bum.
So here's the plan. I bore the hole in the arm with a regular 5/8" Jennings bit. The I thread through the ABE with the 1/2" bit chucked in from below, then I just start the 1/2" bit enough to give the lead screw the angle. Remove the arm and follow up with the 5/8" again, hoping I can keep the angle right. I thought about putting out a plea for a 5/8" improved centre bit, but a) this is getting ridiculous, and b) as the bits get bigger, they also get longer... I even considered getting a brand new spoon bit if it'd do the job, but measurement from a picture suggests it's miles too long.
All I need now is a day when I don't feel like death warmed up and a gallon or two of courage and I'm good to go. You'll know when 'cos I'll either be so elated I'll be unbearable - or so depressed, well, ditto...!