Raymond, A 6-32 tap is the weakest tap that you can use(diameter to pitch ratio), In the material that you are tapping and the depth of hole a 60 to 70 percent ( No.33 and No. 35 drill respectively) thread will work just fine, the threads will still be strong enough to twist off the head of a screw before you pull out the threads. and the hole will be much,much easier to tap. Oh, and be sure to use a quality tapping fluid, like Tapmagic. One more thing, give the hole a small chamfer equal to at least the dia of the tap before beginning to tap.
In my 30 years + of designing machinery I have learned to avoid the 6-32 tap. for the exact reasons that you are having.
A 3mm metric screw is much strong that a 6-32 in that size range,
everything that you need can be purchased from MSC.
Well have fun
Jerry