Which interpretation is correct?

In this sentence, is the “nattya” short for “natte wa” or “natte simatta”?

”先生、そうお怒りになっちゃ、おからだにさわります。”

It’s ‘natte wa’. ~ては becomes ~ちゃ, which is what we have here. The -shimau contraction is ~てしまう to ~ちゃう (which inflects like any verb ending in う, none of which will get you to a bare ちゃ).

Also, なってしまった (or なっちゃった) in particular here won’t work because it isn’t something the second half of the sentence can follow on from.

PS: I suspect you have a typo there and it is actually そうお怒りになっちゃ (with an お) for the keigo o+masu-stem+naru pattern.

Thank you so much for your response!

Yes, there was a typo. I fixed it. Thanks for pointing it out.