First things first: does he understand this is a problem?
If he doesn't, then your challenge is different.
If this is the case, record some rehearsals and then play them back, pointing out the trouble areas, if he is the kind of guy who can be receptive to constructive criticism.
But he won't ever be "all-in" on fixing the issue if he doesn't really think it's an issue, and just a case of the guitarist bitching about nothing.
If he does realize it is an issue, then you are halfway there. Having played drums to a click for well over a decade, I can tell you it is just like every other skill I learned on the drums: I had to actually practice it. The excuse that a decent drummer "can't" do it is just that, an excuse.
I hear what you're saying about "sucking the fun" out of music, but often that is just a rationalization for someone who doesn't want to do something.
The click tracks I designed have a clave on 1,2,3&4, a tambourine on 1&3, and a light cowbell on 1, all at different volumes. It makes for a much more musical click and gives the drummer a clear understanding of where he is at every point in the click. Try it! Or come up with your own version of click musicality. Have him play to a full percussion track (congas, bongos, bells, etc.). Whatever works.
And do your best to explain to him how both you and he don't want to be listening to tracks five years from now and be frustrated by how much the tempo varies. Once those drum tracks are done, they live on forever, good or bad.