Barely set-valued tableaux are a variant of Young tableaux in which one box contains two numbers as its entry. It has recently been discovered that there are product formulas enumerating certain classes of barely set-valued tableaux. We give some $q$-analogs of these product formulas by introducing a version of major index for these tableaux. We also give product formulas and $q$-analogs for barely set-valued plane partitions. Many of the results are stated in the generality of $P$-partitions that then specialize to particularly nice formulas for rectangles and minuscule posets. The proofs use several probability distributions on the set of order ideals of a poset, depending on the real parameter $q>0$, which we think could be of independent interest.