Do you see why a 2-place beauty would be more relevant than a 1-place beauty?

Relevant to what?

I would have no objection to a one-place beautyₐ, where 'beautyₐ' is an exhaustively physically specifiable idea like 'producing feelings of net aesthetic pleasure when encountered by most human beings'. I would also have no objection to a two-place beauty₂, where 'beauty₂' means 'aesthetically appealing to some person X.' Neither one of these is more logically legitimate than the other, and neither one is less logically legitimate than the other. The only reason we prefer beauty₂ over beautyₐ is that it's (a) more user-friendly to calculate, or that it's (b) a more plausible candidate for what ordinary English language users mean when they say the word 'beauty.'

What I want to see is an argument for precisely what the analogous property 'fairness₂' would look like, and why this is a more useful or more semantically plausible candidate for our word 'fairness' than a one-place 'fairnessₐ' would be. Otherwise your argument will just as easily make 'plus' three-place ('addition-according-to-someone') or 'bird' two-place ('being-a-bird-according-to-someone'). This is not only impractical, but dangerous, since it confuses us into thinking that what we want when we speak of 'objectivity' is not specificity, but merely making explicit reference to some subject. As though mathematics would become more 'objective,' and not less, if we were to relativize it to a specific mathematician or community of mathematicians.

"most or all piles of pebbles will have prime membership."

So is your worry that having a one-place 'fairness' predicate will make people think that most situations are fair, or that there's a physically real fundamental law of karma promoting fairness?

Generally?

In general, yes, generally.