Need 32-bit mixing function that has perfect avalanche between octets
At first glance, the MixColumn step from AES (actually, a single column of that transform) sounds like precisely what you're looking for.  It is invertable (AES depends on that), and it does have the property that if one input octet changes, then all four output octets are guaranteed to change.

Most commonly, it's done by table lookup; however there's no reason it couldn't be done by inline code (and that inline code ought to fit in 256 bytes).