This paper studies a fluid-antenna-enabled integrated bistatic sensing and backscatter communication system for future networks where connectivity, power delivery, and environmental awareness are jointly supported by the same infrastructure. A multi-antenna base station (BS) with transmitting fluid antennas serves downlink users, energizes passive tags, and illuminates radar targets, while a spatially separated multi-antenna reader decodes tag backscatter and processes radar echoes to avoid the strong self-interference that would otherwise obscure weak returns at the BS. The coexistence of tags and targets, however, induces severe near--far disparities and multi-signal interference, which can be mitigated by fluid antennas through additional spatial degrees of freedom that reshape the multi-hop channels. We formulate a transmit-power minimization problem that jointly optimizes the BS information beamformers, sensing covariance matrix, reader receive beamformers, tag reflection coefficients, and fluid-antenna (FA) positions under heterogeneous quality of service constraints for communication, backscatter, and sensing, as well as energy-harvesting and FA geometry requirements. To tackle the resulting non-convex problem, we develop an alternating-optimization block-coordinate framework that solves four tractable subproblems using semidefinite relaxation, majorization--minimization, and successive convex approximation. Numerical results show consistent transmit-power savings over fixed-position antennas and zero-forcing baselines, achieving about 13.7% and 54.5% reductions, respectively.