Field-theoretic analysis has predicted that the edge of a double quantum spin Hall insulator (DQSHI) can undergo an interaction-driven transition from a phase with fermionic edge modes into one with only bosonic modes. All single-fermion edge excitations are gapped in this bosonic edge phase, but fermion pairs remain gapless, leading to distinct experimental signatures. We numerically simulate the interacting DQSHI edge and verify the existence of the bosonic edge phase. This calculation requires an explicit bulk embedding of the DQSHI edge, due to its mixed anomaly between charge and Sᶻ conservation. We hence employ segment DMRG on an infinite two-dimensional cylinder, where the DQSHI bulk is modeled in terms of two copies of the spin-up lowest Landau level, combined with their time-reversed counterparts. We find that interactions can open a single-fermion gap at the DQSHI edge while fermion pairs remain gapless, matching the predictions for the bosonic edge phase. We find the scaling of the single-fermion gap with interaction strength to be consistent with field-theoretic calculations, and discuss the relevance of the bosonic edge phase to twisted homobilayers of WSe₂ and MoTe₂. We elaborate on the methodological extension of the segment geometry used here to other topological phases with edge anomalies.