Low-Earth-orbit (LEO) inter-satellite links must cope with strongly doubly selective channels and aged channel state information (CSI). In this paper, the term ``sensing'' refers to the receiver-side identifiability of a small set of dominant delay--Doppler path parameters, quantified via CRLB-type proxies, rather than a full-fledged target-sensing pipeline. Affine frequency division multiplexing (AFDM) provides a sparse delay--Doppler (DD) representation well suited to such channels, yet most existing AFDM designs assume ideal CSI, operate on grid-based channel coefficients, and optimize only communication performance. This paper proposes a two-stage AFDM-based ISAC framework for mobile LEO ISLs that explicitly operates under predicted CSI. In Stage~I, we model the channel by a small number of dominant specular paths and perform sequence prediction directly on their complex gains, delays, and Dopplers, from which we reconstruct the AFDM DD-domain kernel used as the sole instantaneous CSI at the transmitter. In Stage~II, we design a sensing-aware AFDM pre-equalizer by augmenting the classical minimum mean-square error (MMSE) solution with a term obtained from Cramér--Rao-type sensitivity measures evaluated under the predicted channel model, leading to a first-order surrogate of a CRLB-regularized pre-equalizer with a single tuning parameter that controls the communication--sensing tradeoff. Simulation results for representative LEO ISL trajectories show that the proposed path-level predictor improves effective-kernel reconstruction over AFDM-unaware baselines, and that, under predicted CSI, the sensing-aware pre-equalizer significantly improves sensing-oriented metrics over outdated-CSI baselines while keeping symbol error rates close to a communication-oriented MMSE design with only modest additional complexity.