We analyze the critical exponents relating to the quark mass anomalous dimension and $\ensuremath{\beta}$-function at the Banks-Zaks fixed point in quantum chromodynamics in a variety of representations for the quark in the momentum subtraction schemes of Celmaster and Gonsalves. For a specific range of values of the number of quark flavors, estimates of the exponents appear to be scheme independent. Using the recent five-loop modified minimal subtraction scheme, quark mass anomalous dimension, and estimates of the fixed point location, we estimate the associated exponent as 0.263--0.268 for the $SU(3)$ color group and 12 flavors when the quarks are in the fundamental representation.