We explore different implementations of the quantum Bell causality (QBC) condition in the quantum sequential growth (QSG) dynamics of causal set quantum gravity, for non-commuting transition operators. Assuming a non-singular dynamics we show that for the two most natural choices of operator orderings for the QBC, the transition operator algebra $𝒜$ reduces to a commutative one. As a third choice, we take the operator ordering to depend on the size of the precursor set. We find several new commutation relations which further constrain $𝒜$ but do not imply commutativity. On the other hand, if any of the generators of the "antichain subalgebra" $𝒬$ belongs to its center, then this implies commutativity of $𝒜$. The complexity of the algebra prevents us from obtaining a general form for the transition operators, which hinders computability. { In an attempt to construct the simplest non-trivial $d=2$ representation of $𝒜$, we find that a Pauli matrix representation of the generators of $𝒬$ leads to inconsistencies. Our work can be viewed as a first step towards finding a non-commutative realisation of QSG.