Personally I despise Tiny Hut because long rests are one of the most broken things about D&D. It's already hard enough to balance encounters without this spell.
Instead of banning it, I had a wild idea: what if it was a concentration spell? Meaning whoever is casting it has to stay up and...
Since long rests including healing up to full HP, if any damage is taken during the long rest, the long rest is cancelled and must be restarted.
As for breaking up sleep, sleeping in 4 hour segments is acceptable due to standard watch periods, but it also assumes that your watch is 4 hours long...