The tripartite quantum Rabi model couples two qubits to a bosonic mode through a collective spin-oscillator interaction, providing a simple setting for studying two-qubit entanglement. In the zero-detuning limit, the triplet part of the spectrum splits into branches with zero and maximal entanglement, while the antisymmetric singlet ladder remains exactly decoupled. Within the triplet sector, finite detuning turns the crossings between these branches into avoided crossings and redistributes this entanglement. We identify an eigenbranch whose entanglement grows from nearly zero to a nearly maximal value through such avoided-crossing mixing. The weak-coupling level ordering yields a simple analytic criterion for whether this eigenbranch has a separable weak-coupling endpoint. A three-state effective model explains how the Bell-state component becomes dominant as the coupling increases. We further compare linear and local adiabatic ramps of the collective coupling under common constraints, measuring the time needed to prepare a Bell-like state on the selected eigenbranch. These results show how collective spin-oscillator coupling reorganizes spectral entanglement and connects static branch structure to finite-time Bell-state preparation.