Any process that modulates the rate, frequency, or extent of the Wnt receptor signaling pathway through beta-catenin involved in neural crest cell differentiation. The Wnt receptor signaling pathway is the series of molecular signals initiated by binding of a Wnt protein to a frizzled family receptor on the surface of the target cell, followed by propagation of the signal via beta-catenin, and ending with a change in transcription of target genes.