John Doe wrote:

snip

I need more strength than just enough to thread the machine screw
through a correctly tapped and cleaned hole. If the screw is going
to break so easily, it probably won't hold the pieces together
either. That's my problem.

You shouldn't be breaking even the poorest screws just threading them
into a tapped hole without tightening them. Something is seriously wrong
here. Either that or you're trolling :-).

Try McMaster-Carr. They may have sets of screws.

Best wishes,

Chris