Well, it took me 3 months, but I've got some answers.
(Yes, how shameless of me to answer my own question.)
Here are some voicings I found that satisfy my original qualifications (keeping Root, 3rd, 7th, and highest extension, not too painful). These, of course, can be moved up and down the fretboard to different roots.
- 5-3-6-5 (C9)
- 4-5-3-7 (G13) (also 4-5-5-7 to include the 9th)
- 3-4-5-3 (C13) (also 3-4-5-5 to include the 9th)
- 7-4-7-9 (E9 or Em9) (yes, I know this doesn;tdoesn't totally fit and it's a stretch, no pun intended)
- 3-2-3-3 Gm(Gmadd1111)
- 4-3-5-5 B7♯9
- 7-4-7-8 E7♭9
- 2-4-1-3 G13
- 3-2-5-3 B♭maj9
- 7-6-9-11 E13
- 3-1-4-3 D♭6/7D♭maj13 or B♭m9
- 7-6-8-6 D7♭9
- 3-7-5-6 E♭(♯11)
- 2-3-3-1 E♭(♯11)
- 5-1-6-6 D♭maj13
- 4-1-6-6 D♭13
- 7-4-5-9 E9sus
- 7-9-7-10 A9sus
- 4-5-6-6 G7♯9♭13 (or, to trade the ♭13/♯5 for the ♯11, 4-5-6-4)
- 3-4-4-6 C7♯9♭13 (or, to trade the ♭13/♯5 for the ♯11, 3-4-2-6)
I've by no means exhausted the list of voicings for 9ths, 11ths, and 13ths, so if anyone finds more, please let me know! :)