bi-metal should do it fine, slowish and oil WATER will help, as would a drill press if possible

Fixed it for you.  When cutting/drilling you want to use a coolant, not a lubricant.  Oil does not dissipate enough heat and will make it get hot when drilling thicker pieces.

For the 0.060 steel I would just drill it dry, won't take much longer than a minute drilling 16ga.  I have a 15/16" bi-metal hole saw that I use to drill well pipe when building free standing panels that has built several panels with 40+ holes per panel and still cuts like new and only ever drilled with water and those a closer to 1/4" wall.  A drilling coolant would be better since is has a rust inhibitor in it, but the panels are never painted so it doesn't bother me if they get rusty.