Is the analysis in the story correct, and is it permissible to use (Walah) as part of a pseudonym, considering that the name means "demon of the privy"?

There is no knowledge about analyzing the mentioned story. There is a devil for ablution called Al-Walhan, but there is no devil for the toilet called Walh. The name may affect the named, so the Prophet, peace and blessings be upon him, changed some names, and whoever named his son after a devil should change it.