Cat-state　qubits　formed　by　photonic　cat　states　have　a　biased　noise　channel,　i.e.,　one　type　of　error　dominates　over　all　the　others.　We　demonstrate　that　such　biased-noise　qubits　are　also　promising　for　error-tolerant　simulations　of　the　quantum　Rabi　model　(and　its　varieties)　by　coupling　a　cat-state　qubit　to　an　optical　cavity.　Using　the　cat-state　qubit　can　effectively　enhance　the　counterrotating　coupling,　allowing　us　to　explore　several　fascinating　quantum　phenomena　relying　on　the　counterrotating　interaction.　Moreover,　another　benefit　from　biased-noise　cat　qubits　is　that　the　two　main　error　channels　(frequency　and　amplitude　mismatches)　are　both　exponentially　suppressed.　Therefore,　the　simulation　protocols　are　robust　against　parameter　errors　of　the　parametric　drive　that　determines　the　projection　subspace.　We　analyze　three　examples:　(i)　collapse　and　revivals　of　quantum　states;　(ii)　hidden　symmetry　and　tunneling　dynamics;　and　(iii)　pair-cat-code　computation.