I'm a bit curious what the stuff is that they do want to play. "Alternative" can mean a lot of different things, and some crowds like "heavy." Taking the OP's word that they want obscure dance-floor-killing stuff, I'm on his side, but it would be nice to see set lists.
I've always felt that the band itself is one of the constituencies you need to keep happy for the band to keep working. My rule is that the set lists should be built for popularity, but each member of the band can get a "vanity" pick - a song to play just because they want to play it. Which, in a 4- or 5-piece group, would yield the 4 deep cuts you mention, out of say 40 for the night. And insert them where they won't clear a floor, like early in the 1st set while people are still getting warmed up or late in the 3rd.
Attempt to get the to only substite about 4 deep cuts for the same amount of songs you already know people enjoy by their dancing, and they'll be able to see what goes over better without having to go all or nothing on overhauling the set with them.
Otherwise I like the idea that was already mentioned above ^ about starting another band with them for that obscure stuff.
I've always felt that the band itself is one of the constituencies you need to keep happy for the band to keep working. My rule is that the set lists should be built for popularity, but each member of the band can get a "vanity" pick - a song to play just because they want to play it. Which, in a 4- or 5-piece group, would yield the 4 deep cuts you mention, out of say 40 for the night. And insert them where they won't clear a floor, like early in the 1st set while people are still getting warmed up or late in the 3rd.