You won't get any better result with BMP, just a lower compression rate...
The PNG file just stores the colors you create, so if you want results closer of what you want, you have to act on the sketch, not on save format.
Perhaps just deactivate smooth(), if you have it.
You can also act on the array of pixels to quantize the colors.
Etc.
Hard to give a precise answer without seeing code. I am not sure what you call "half-tones", and why you use 8 bits for 3 colors...

Note: you can also do post-processing (no pun intended) of the resulting image, ie. using an image software to change the format. PNG supports 8 bits per pixel, too.

Thanks! I think I'll go with turning off the smoothing and combine it with "pixel perfect" .png graphics that I import and mask.