An efficient, flat histogram Monte Carlo algorithm is proposed that simulates long-range spin models in the multicanonical ensemble with very low dynamic exponents and drastically reduced computational effort. The method combines a random-walk in energy space with cluster updates, where bond weights depend continuously on the lattice energy. Application to $q$-state Potts chains with power-law decaying interactions is considered. Lattice sizes as high as $2^{16}$ spins, unattainable with conventional flat histogram algorithms, are investigated. Numerical results demonstrate the remarkable performance of the method over a wide spectrum of model parameters.