To partially answer my own question:
The problem with the conversion software is caused by the fact that these files have multiple palettes. So I need to find (or write) conversion software that can handle multi-palette .tex files. Can anyone point me to something?
I was wrong about Reunion R06 ignoring some .png textures. What's really going on is that the Enhanced Stock UI mod I was borrowing from doesn't have .png textures for the menu font because it uses hext files to make the menu use the field font. Unfortunately, that hext isn't fully compatible with Reunion R06, so I'm going to have to make new font textures. But at least it can be done with .png and I don't need to convert back to .tex.
Which is great because, as per
this thread, it's not possible to upscale the font .tex files because sometimes the .exe grabs rectangles from them using fixed integer offsets and dimensions (rather than offsets and dimensions expressed as fractions of the texture dimensions).