Whether or not Brims has menace matters only as blockers are declared. After that point, a gust of wind removing your hat or blowing a new hat onto your head won’t change which blockers have already legally been declared.
When "Brims" Barone, Midway Mobster enters, put a +1/+1 counter on each other creature you control that has a hat.
"Brims" Barone, Midway Mobster has menace as long as you're wearing a hat.