Vocab problem: 肥 vs 胖 vs 肥胖 Do all of these mean just "fat, obese"? Dictionaries seem to indicate that these characters mean that separately. Is there a reason to use them both? Is this one of the cases where two characters just remove ambiguity?
Is this one of the cases where two characters just remove ambiguity?

Yes, Since each Chinese character has its own meaning or meanings. A single character, usually cannot clearly express a specific meaning, therefore two characters would combine into a specific word, and two words may combine into one specific phase.(see my other post How to determine if character can be used separetely?)
In the case of 肥 and 胖. both can be an adjective for &quot;fat&quot;, however 肥 also has other meanings:

肥 http://www.cantonese.sheik.co.uk/dictionary/characters/188/
[1] [adj] fat; plump; chubby | [antonym] 瘦
[2] [adj] greasy; oily; fatty
[3] [adj] fertile; rich; luxuriant
[4] [n] fertilizer; manure
[5] [adj] profitable; lucrative; affluent

Combine 肥 and 胖 into 肥胖 make it a specific term for &quot;fat /obese &quot;. Since it is a specific term, it is also more formal.
Beside being an adjective for &quot;fat&quot;,  肥胖 can also be a noun for &quot;obesity&quot;