Carter Picking is a style where you play the melody in the bass with your thumb (usually the E, A and D strings of the guitar) and brush the treble strings with your fingers to fill out the chord and establish rhythm. Some players do it with a flat pick, but usually the style is played fingerstyle, with or without a thumb pick. Youtube 'Mike Seeger Carter Style' for an awesome demo from a true master.
Travis style is also fingerstyle, but completely different. The melody is played in the treble with the fingers, and the thumb plays a (usually) octave alternating bass, which is damped with the heel of the hand for a boom-chuck rhythm. Youtube Merle Travis to see travis style. Or just 'thumb style guitar.'
Both are great guitar styles but tough to adapt to the uke due to the lack of bass strings. Even with the low G, you need lower notes, to get the melody notes for Carter style, and to get the octave for the alternating bass for Travis. Otherwise your bass notes are overlapping your treble. I think of those styles as basically needing six strings.
But if anyone can adapt those styles to the uke, it would be Aaron Keim. It would be an adaptation, maybe giving something of the feel of Travis or Carter. To me, both styles need a deep, loud bass, which the uke hasn't got, even with the low G.
Oh, and I don't see a clear connection between either of those styles and clawhammer, which is a banjo style developed to accompany a fiddler (bum-ditty) while Carter and Travis are guitar styles (boom-chick) -- 3 vs 2. Travis is really derived from ragtime piano. They are all mountain styles, however...