This is a mistake requiring the use of formal grammar.

the use of "Let ... come in" requires either a noun or a pronoun (ie one of these grammatical words that replaces a noun -eg when you want to avoid being redundant)

The issue with "his" is that it is not a pronoun such as me/you/him/her/it/us/they (and generally it is not a noun) in English, but, rather, a possessive adjective (as in "his book") or a possessive pronoun (replacing a possessive adjective and the noun that follows...).
Therefore 'Let his cat come in' can work, and then on its own 'don't let my cat in, but let his come in' could possibly work as an exception, though it is very heavy...