I missed the part where the manager said or did something to insult your intelligence.

I suspect there's some important detail missing from the letter, like the manager saying "yeah, like I really believe that another store is selling this mirror for less."

On the other hand, your insistence that his inability to price match wasn't true... that's insulting him. If you had proof that he was able to do so, you should have raised that to his attention. Ditto on your insistence that he leave the store and walk over to a competitors store.

Maybe it was stupid of him not to price match, but it's not patronizing to say "no."