[Imladris]

I have always viewed it this way:

If my memory serves me truely, the Black Riders were asking for a Baggins, correct? Well, the hobbits knew that because of what Frodo had overheard before he left Bag End. Fredegar's disguise wasn't for the Black Riders, but for any hobbits who happened to see him in a window or something. Even though Fatty had been instructed to tell any visiting hobbits that Frodo was busy or sick, that doesn't mean that they might accidentally see him through a window. That way, when any Black Riders came and asked where Baggins was, the hobbits would say he was at Crickhollow.