This paper explores high-altitude platform station (HAPS) systems enabled by integrated sensing and communication (ISAC), in which a HAPS simultaneously transmits communication signals and synthetic aperture radar (SAR) imaging signals to support multi-user communication while performing ground target sensing. Taking into account the operational characteristics of SAR imaging, we consider two HAPS deployment strategies: (i) a quasi-stationary HAPS that remains fixed at an optimized location during SAR operation, following the stop-and-go scanning model; and (ii) a dynamic HAPS that continuously adjusts its flight trajectory along a circular path. For each strategy, we aim at maximizing the weighted sum-rate throughput for communication users while ensuring that SAR imaging requirements, such as beampattern gain and signal-to-noise ratio (SNR), are satisfied. This is achieved by jointly optimizing the HAPS deployment strategy, i.e., its placement or trajectory, along with three-dimensional (3D) transmit beamforming, under practical constraints including transmit power limits, energy consumption, and flight dynamics. Nevertheless, the formulated optimization problems corresponding to the two deployment strategies are inherently non-convex. To address the issue, we propose efficient algorithms that leverage both convex and non-convex optimization techniques to obtain high-quality suboptimal solutions. Numerical results demonstrate the effectiveness and advantages of the proposed approaches over benchmark schemes.