However I don't see why PSP should not be able to handle the job.

It turns out that it's because PSP can't handle 32-bpp raster images. According to JASC support:

Paint Shop Pro does not support 32-bit images. Hence it will not support the format that you are trying to create.

More specifically, it'll do all the layering and stuff while editing the file, but when you go to save it to a BMP or PNG file, it flattens everything down to 24 bits, and strips out the alpha channel.

This means that I can't use it to save a 32-bpp PNG file (because it doesn't actually seem to write them) even if I spend the time to write a PNG to BMP convertor (which should be trivial, given libpng).

So, frankly, I think I'll opt for manual pixel pushing -- editing the 24bpp image and the alpha separately, and then writing a utility to merge them back together. This way I get to use the copy of PSP5 that I've had for years, and I don't need to spend any more money.