for such a small size you need a perfect concentricityin your lathe setup, a good light, and a magnifier.
I did it exactly for the same aim, butane burner jets, so it's possible.
0.2 mm drills can be found in the trade but pretty expensive and do not expect to drill more than a length of 0.5 mm. 0.3 mm are easy to find, much cheaper, and can be reduced as follows.
the drill must not be hold directly in the drill chuck or collet but in a home made collet : a small piece of brass is turned to 3 mm dia for a max of 15 mm, then drilled concentrically half-way 1.5 mm dia then 1 mm for half of the remaining length and drilled through with the 0.3 mm drill, and finally cut. This is your collet, you may either slit it longitudinally or simply use loctite to hold the drill. insert the drill in the collet leaving only a minute amount of it, no more than 1.0 mm.
You can use it as such or with your dremel and a small grinding wheel (dressed with a diamond stone) in a toolholder you can carefully reduce the drill diameter to suit your need.
Drill break occurs when the workpiece is not dead flat or the drill is not properly centered on the workpiece, so do not use a center drill, but make one : a small sharp coned D-bit will do it.
when your are at it with the dremel on the tool post, try to make your own tiny flat drill with the tail of a broken drill or 2.3 mm dental burr : grind a cone about 90°, then reduce the dia to the wanted size for a 1 mm length, and with the lathe stopped the awkward part is to mill 2 flats 180° leaving some thickness between. these drills work very well, but are very fragile, and break easily on storage.
