As stated above, spin it as fast as you can otherwise the surface speed of the bit is too slow to do much. The most likely cause of a broken bit is that the chips got stuck in the hole so, yes, peck and don't try to cheat; just do it. In my experience Dremel tools don't have wonderful bearings, so if the hole size is real important, I'd not go that way. Buy several drills. In a previous life we found that some of the tiny ones look like drills but don't actually cut. Also it's likely you will break one or two. Since you have a lot of time in the part already, I'd be inclined to learn how to use the tiny drills on a test piece before doing it for real. In fact, I find that learning how to do a new process is almost always worth doing on a trial piece to get down the learning curve a bit before trying it for real. A Dumore high speed, precision drill would be a great tool for this, but a bit spendy for just a few holes.
Good luck!!