With 2433 you don't have to actually barre with the index finger: it only has to stop the 4th string. You just want to partial-barre the 3's with your middle finger (or you could use your ring finger and fret the 4 with your pinky). You might be able to partial-barre without collapsing the first joint, by fretting both strings with the ball of your finger—lower your wrist a little to get a wider area of contact, and center your finger between the strings (this is how a fiddler plays fifths). Or you can use an angled index barre, stopping with your index both the 4th string at fret 2 and the 1st string at fret 3, using your middle finger to stop the 2nd string only. This is a position of last resort, since it forces you to tuck in your elbow and angle your hand in the opposite direction that you usually angle your hand when you're not playing perpendicular to the strings—lots of extra motion moving into and out of that position.
Still, 2433 (n:0211) is a common m7 shape when playing up the neck, so it pays to master it somehow—you can't use the n:0453 shape higher up unless you're pretty limber (though for movable shape n:0211 you might be able to substitute the full barre shape two frets lower, just as 0000 substitutes for 2433).
As a solid chord (no glissando or arpeggiated roll), 0453 sounds exactly like 2433 on reentrant ukes, as does 5430; even glissed or rolled, it probably won't make a bit of difference. With serial tuning, all three still play the same chord but with different character. I mainly prefer 2433 because I tend to think in terms of movable chord sequences rather than absolute chords, and like to be able to shift the base key with minimal change to hand patterns. As NewKid pointed out, muting/cutting off chords is easiest with movable shapes: you just relax your left hand pressure. Of course, if you finger 0453 or 5430 with your first three fingers, you can still left-hand mute the chord with your pinky, but if 2433 causes you grief, you're probably not much into movable chords and muting isn't yet a major concern to you; the other shapes will do fine.
Although C#6 is not an equivalent of Am7 and 1423 (Bbaug7) doesn't serve for any of these chords, DownUpDick was attempting to point out that Am7 and C6 share the same set of pitches. The same full-chord formation works for both, whether it's 0000, 2433, 4530, 5430 or a formation higher on the neck like 5757.
