Follow up, I still haven't found the ideal solution, which is pixel-perfect scaling.
Changing the sprite filtering to smooth instead of pixelated stops the distortion, but it's also very blurry looking. With pixel art, this looks like hot garbage, but if you use higher resolution assets, it's fine.
Also, when the sprite's transparency is set to Opaque, it looks really bad, but it's fine with transparency on.
But it looks pretty okay and doesn't have the ugly distortion issue, so... I guess it works?
My conclusion: When it comes to Superpowers, 256x256 is 8 times better than 32x32.