Recent work (Nathan , {PRX Q {}, 30352}) proposed an architecture for a dissipatively corrected GKP qubit, and protocols for self-stabilizing, exponentially robust Clifford gates. Here we propose a protocol for a self-stabilizing, exponentially robust non-Clifford $\sqrt{T}$ gate for the qubit based on the inclusion of a quartic flux potential generated by ancillary Josephson junctions. We show that gate infidelity is exponentially suppressed in the presence of control or device imperfections, and that the gate operates on microsecond timescales for GHz resonators. The protocol can straightforwardly be applied twice to yield an exponentially robust $T$ gate which could be used as a resource for further distillation. We analyze the resilience of the protocol to noise, imperfect control, and imperfect targeting of circuit parameters.