Frodo's clothes at Crickhollow

[The Only Real Estel]

Imladris, are you supposing that the local Hobbits would be deceived into thinking that Fredegar was Frodo simply by virtue of his wearing Frodo's clothes

I think that's what Tolkien was implying. Even if they weren't deceived by it, you have to realize that it's only a minor hobbit addition to the masterplan, hardly fool proof, & as simple as the hobbits themselves.