I would check the string slots depth in the nut, the action at the 12th, and head tension...then set the bridge. Sometimes if you tighten the capo too much,and the action is too high at either end, the string deflection will cause intonation problems.
I think nearly all capos are, by simple physics, going to throw the banjo "out of whack," even if ever so slightly. It is, after all, pulling the strings down to the/a fret; and then you're fretting strings at different spots, so what is happening is the string is being pulled down even further. It does take some practice getting used to a capo and positioning it just right, so that it doesn't, any more than is practical, pull the banjo out of tune or out of whack any more than absolutely necessary. I have found that non adjustable type capos, (the heavy duty spring type comes to mind) really do pull the instrument out of tune worse than one that can be adjusted - and even those generally take a bit of getting used to. ONLY tighten it down as much as is needed to put the string against the fret with no buzzing, and it also needs to be properly spaced behind the fret, not too close nor too far. Even the best capos, at least as far as I have discovered, generally require just a bit of tweaking your tuning, to get all things as close as possible, or at the very least, to have the banjo NOT sound completely out of tune.... it is NOT a magical thing nor any mystery involved!! If you really have tried some good capos, and still just have terribly "out of tune" things going on, then I suspect your set up is totally off, OR you are need of some serious attention to your banjo!!
If intonation is ok when you are fretting with your fingers rather than a capo then it's the capo, or the way you are using it, that's at fault. If not then it's the banjo (assuming you are fretting correctly). Capo-ing is just fretting by other means. I use a basic Shubb just tight enough to do the job and just behind the fret. Works for me.
Before I begin playing with a capo, I play a few notes and make sure the capo is working correctly. For me using a capo on my guitar is "trickier" than it is on the banjo. You might check your fingerboard and frets out to make sure everything is correct.
When you fit a capo and then fret a note you're effectively depressing the string twice, thereby stretching it more than normal. This happens on guitars as well, 12-string guitars being even worse! I've always accepted this to be a fact of life and simply retune the instrument so's it's in tune when the capo is fitted. After that it's proceed as normal, though the action may well be slightly lower than you're used to ;)