I think it would be a good idea to watch the Z axis depth while allowing the tap to 'cut air' at whatever speed you are experimenting with. If the machine coasts an extra mm before it reverses, then you have to allow for this with your drilling depth and with the commanded tapping depth.
I like to grind off the pointy end of the small taps, just to gain that extra safety space.
Tapping in high gear is probably better too, because the spindle motor won't be spinning as fast. This makes a noticeable difference on a Haas, in how quick the tap reversal occurs.
If you cannot find spiral flute taps in that tiny size, you may end up using 'plain taps' and these are sensitive to chip buildup. You might need to step tap the hole, just to permit lubricant in properly, and the chips out. For comparison, I tap a lot of blind #0-80 holes about 4 to 5 mm deep and always step tap. Hand tap a few holes to gain a feel for how well the tap is cutting, and how far you dare go before backing up.
First you get good, then you get fast. Then grouchiness sets in.
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)