Bayesian calibration of a coupled-channel reaction model requires many expensive solves, and the likelihood gradients needed by Hamiltonian Monte Carlo are absent from most production solvers. I present DREAM (Differentiable Reduced-basis Emulator with Automatic gradients for Markov-chain inference), a strategy that makes a reduced model differentiable without differentiating the original solver: parameter-dependent operators and solution snapshots are exported offline, compressed by singular value decomposition, and reconstructed with differentiable operations, so each likelihood evaluation returns the observables together with the exact gradient of the emulated likelihood. The construction applies to discretized linear models whose parameter dependence admits a low-rank, low-dimensional operator representation, and direct solver calculations verify the reduced model at the posterior points where it is used. Applied to continuum-discretized coupled-channels calculations of $d$+${}^{58}$Ni elastic scattering at $20$~MeV, with eighteen central-potential parameters carried by twelve sampled coordinates, four chains yield 16,000 retained samples in under an hour on one graphics processor, with an offline investment equivalent to about one thousand direct solves. Within the stated reaction model, continuum discretization, prior, and error assignment, the data determine the summed real and imaginary volume integrals as $772\pm 13$ and $204\pm 14$~MeV,fm${}^3$: the real integral lies $15\%$ below the summed Koning-Delaroche reference and is stable under a global factor-of-two rescaling of the assigned experimental errors, while the proton-neutron decomposition of each sum is left undetermined.