Have you tried it by now? I find this a really interesting idea, and would expect it to sound not weird but just very fat, powerchord-ey – provided you use B and F♯ for the second course; a tritone would by tricky indeed...:
E2,B2 A2,E3 D3,A3 G3,D4 B3,F♯4 E4,E4
as Bob Broadley said, this would probably have the effect of organ stop cascading.
Some tunings that might also work well:
B3,E2 E4,A2 D4,D3 G4,G3 B3,B3 E4,E4
This is almost standard 12-string tuning, except the two low strings are now doubled with tritaves instead of octaves. Which are overtones proper, so this tuning should never cause any harmonic problems. Could also be done with the low string of the two lowest coursest replaced:
E3,B3 A3,E4 D4,D3 G4,G3 B3,B3 E4,E4
This might give you a very silky sound that works in rich mixes without ever getting boomy, while still retaining some illusion of having bass strings, albeit with missing fundamental.
E4,E2 E4,A2 D4,D3 D4,G3 B3,B3 E3,E4
This one has the sympathetic strings going down from 6th to 1st course. As a result, there isn't really an orientation of high strings vs low strings anymore, but you can still play chords and, to some degree, melodies like you're used to. D-shape chords will sound very rich in this tuning, not so unusually lean as they do with standard tuning because only using the upper four strings.
E4,G♯4 A3,E4 D4,A4 G4,G3 B3,B3 E4,E4
This uses for every string the highest combination which only contains overtones of the standard-tuning fundamentals. Make sure the G♯ forms a just major-third, not a 12-edo third. This tuning will sound very light and thin, probably not too useful on its own but might be extraordinarily effectful for doubling a 6-string guitar.
E3,G♯3 A3,C♯4 D4,F♯4 G4,B3 B3,D♯4 E4,G♯4
It's getting a bit more experimental here. This is the same general idea as the all-fifths tuning, but with thirds instead (it's paramount that these are all in just intonation). Will give a very peculiar sound, but it should still be musically comprehensible if you're playing standard guitar stuff in this tuning.
G♯3,D₇4 E3,G₇4 A3,F♯4 B3,G4 D♯4,F♯4 D₇4,G♯4
Ok, now none of the courses actually contain the note you'd have in standard tuning. Still, provided you tune the harmonic sevenths correctly, they should all be coherent to the standard fundamental. This will sound pretty crazy, but perhaps be actually usable for some stuff. Try chords with capo 7 or something.