We study the entanglement structure of the tripartite quantum Rabi model, in which two qubits couple collectively to a bosonic mode. At zero detuning, the triplet spectrum separates into branches with zero and maximal entanglement while the antisymmetric singlet ladder remains decoupled. Finite detuning turns selected crossings into avoided crossings and redistributes entanglement between eigenbranches. A three-state effective model explains how one branch evolves from a nearly separable weak-coupling state into a state dominated by a Bell component. Finite-time coupling ramps then provide practical requirements for high-fidelity Bell-state preparation.