The name could be dropped in favor of just the icon on smallest breakpoints. This will also leave room for wallet error messages.
It seems likely users would expect clicking the icon and/or name to do something; my suggestion would be to have it go to the DAO's homepage.
Icon size: 48x48px, same height as the connect wallet button
DAO name text size: H4
DAO name should be centered vertically in relation to the icon; the icon should have the same top/bottom and left margins as the wallet connect button does (on the right, in the wallet connect button's case)
It can be confusing when using the Admin app to know which DAO you are currently viewing.
Add DAO name above the navigation bar on desktop and mobile views.