The drill gets hot and needs lots of opportunities to cool down, plus you can’t drill far without bunging the thing up with dust which makes it hard to extract the bit, so it’s best to drill just a couple of centimetres each time, then clear the tube, then put the bit back in and measure with a magnet. That should prevent it from wandering a long way off course as the course corrections can be made soon enough to ensure that there’s always plenty of leeway to make stronger corrections if it keeps going off track due to density differences through the rod. If the head was to reach 2mm away from the centre, at that point there would be no way to steer it back other than relying on luck, or maybe trying to bend the rod, but it should never be allowed to get anywhere near that far out: the magnet can detect much smaller movements to the side and can likely keep the wandering down to under half a millimetre off centre.
Once half way through the drilling and with the head right in the centre, it may also be possible to switch to an auger and let it steer itself the rest of the way as it likely can’t go significantly off course after that.
The only idea I had with regard, is that if tilting the rod to steer the bit is not enough, then a half pipe of say plastic (e.g. 0.7 cm thick plastic if the gap between rod and wood is 0.5cm when centered) could be pushed down the side where the drillbit is too close to edge, so making the drillbit move back towards center by leverage.
That might help in an emergency where it keeps wandering strongly towards the edge, though it could be hard to keep it in the right place with the shaft trying to move it round. Grease on the inner surface of the plastic might fix that and reduce the added friction. The friction does get high, but it may be that after every couple of centimetres bored you can pull the bit back a little (to get the screw-tip out of the pilot hole that it makes for itself), and then run the drill while moving the shaft in circles at the opening of the tube to make the head widen the bore a fraction.
2mm is maybe thick enough if the wood is good etc. , but in my opinion it is about at limit even then. I have made “bores” for lining bores about 1mm thick, and they were solid enough to handle normally (but not to apply great pressure to), but the problem is that if they were the actual bore, moisture changes etc. would probably weaken, warp and split them eventually. One flute ended up with a paper thin (literally) area (about 1cm wide by 3cm long) that I could flex by finger pressure… it lasted for a few months before a very fine split appeared, unnoticeable by eye but enough to make the flute jump octaves randomly…patched it and all good again. For tuning I think there are both advantages and disadvantages to a thin walled bore, but I leave that there.
I’ll use the 16mm bit again the first time, getting 4.5mm thick walls on average, and if that goes well I’ll give an 18mm bit a go: that’ll give me 3.5mm walls which may still be enough.If it wanders off by half a millimetre, that should still be acceptable unless the mismatch between the thicknesses of the walls on opposite sides affects the tone badly. If that doesn’t spoil the sound, it might be possible to put the bore slightly off centre on purpose in order to have slightly thicker wood where the holes are, though the easier solution if I’m going to make more than a few of these would be to buy a lathe and solve all the problems that way while drilling straight holes with augers and not needing to care if they go off in the wrong direction by a few degrees - I see now that high quality lathes that can handle the lengths of instrument I’m making start at about £300, which is a fraction of what I was expecting, and it would give me a better choice of wood to use. I just want to be sure I can make instruments of adequate quality before I start putting that much money into it.

