Hmmm. Well, for the size issue - these are *old* clothes of Frodo's, and it's stated that Frodo has gotten much thinner recently, what with all the walking (probably also with stress). So Frodo may have been fairly plump back in the day, so Fatty would be able to wear his clothes fairly comfortably.
I believe that what Fatty did was simply put on Frodo's clothes from time to time when he was messing around inside the house or in the garden. (A cap would probably go a long way towards obscuring the identity of the wearer). Crickhollow always sounded to me like a somewhat remote place, and Frodo isn't going to be exactly besieged by visitors - probably the only people who would see Fatty would be a few hobbits going by on the road at a bit of a distance, and maybe the butcher and the milkman, assuming they do home deliveries. None of these people are going to know what Frodo looks like, and they're not likely to be calling on him. They also don't know that Fatty is living there, so it's natural to assume that any figure they see is the Mr. Baggins who just moved in. So talking among themselves, word will get out that Frodo Baggins is at Crickhollow and remains there. That's the danger to Fatty right there - if the Black Riders heard that a Bolger was living at Crickhollow, they wouldn't think of it twice. It's the fact that Fatty is masquerading as a Baggins that makes the Black Riders likely to come knocking; and of course, had they gotten hold of him, they would have killed him regardless of his surname. But he would never have been in danger if it weren't for his impersonation.
Now as for why he would need Frodo's old clothes when he would only be seen at a distance - I can only posit that Frodo, being probably one of the wealthiest hobbits around, wore rather more expensive clothes than Fatty possessed, and it would look strange to outsiders if the once-wealthy Mr. Baggins was wearing common clothing. Perhaps Frodo wore brighter colours (being able to afford certain dyes used to say a lot about you) or maybe his cloaks and clothes were made of especially good fabric, but at any rate, people might have started to talk if the supposed Frodo had appeared in a ratty old cloak. They might not have had much communication with Hobbiton, but they'd know that Frodo was rich, and they'd have expectations as to how rich gentlehobbits were supposed to dress.
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Father, dear Father, if you see fit, We'll send my love to college for one year yet
Tie blue ribbons all about his head, To let the ladies know that he's married.
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