Since Mirriam-Webster's thesaurus lists
Transform and Transmute as synonyms the first definitions of each word seem effectively identical.
The same page then goe s on to make the kind of distinction you seem to be making.
Transform may imply a mere changing of outward form or appearance
...
Transmute usually suggests a fundamental change, especially one involving a metamorphosis of a lower element or thing into a higher one
Clearly this definition of transform is not applicable to evolution. The evolutionary genetic changes we see are obvious not merely a change of outward form or appearance, they are a change in the basic genetic structure of the organism.
Once again your definition of transmutation shows it to be inimicable to evolution as it is scientifically understood, not as you suggest required by it. There are no categorically higher and lower things in evolutionary biology.
TTFN,
WK