We examine a transmission problem driven by a degenerate nonlinear operator with a natural interface condition. Two aspects of the problem entail genuine difficulties in the analysis: the absence of representation formulas for the operator and the degenerate nature of the diffusion process. Our arguments circumvent these difficulties and lead to new regularity estimates. We prove the local boundedness of weak solutions and establish an estimate for their gradient in BMO-spaces. The latter implies solutions are of class \( C^{0,\mathrm{Log}−\mathrm{Lip}} \) across the interface.