I believe the followings are okay:
- 信用するでない。 (explained here, somewhat colloquial)
- 信用するなかれ。 (orthodox classical grammar, sounds highly archaic)
- 信用することなかれ。 (orthodox classical grammar)
信用するでなかれ sounds a little odd at least to me, but my knowledge about classical Japanese is limited, and I cannot say it's incorrect with confidence.
EDIT: Of course I'm aware that this 昔話 is not a real archaic document, and the phrase 信用するでなかれ is produced by someone living in the modern era. Unsurprisingly, modern Japanese people (including myself) are not very good at the rules of archaic Japanese, and they sometimes come up with strange expressions.
At any rate, 信用するでなかれ sounds a little weird even as a modern role word, to the ears of a modern Japanese speaker like me. BCCWJ has literally zero example of such
verb + でなかれ, whereas you can easily find lots of
verb + (こと +) なかれ examples such as 言うなかれ, 侮るなかれ, 思い惑うことなかれ, 礼を乱すことなかれ, etc. So it can be safely said that
verb + でなかれ is at least rare even as role language of wise old men, whether or not it's grammatically correct.