But how does this bigger fear explain why he can't enter without an invitation?
###It doesn't. The inability for Dracula to enter a home uninvited is a superstition that he falsely believed in just like all the other superstitions.
A beast [dog] enters a home only when invited, and if the beast attempts to enter unwelcomed then it's subject to being disciplined. The fears/superstitions that Dr. Zoe Van Helsing mentioned at the end of S01E03 were described as, "The Rules of the Beast". Well, that's also the title of S01E01, where Dracula and Agatha Van Helsing talked at the gates of the nunnery and she mocks Dracula for being like a dog/beast.
That's how it ties in with everything.