In Hamilton-Jacobi reachability analysis, reachable sets are characterised by sublevel sets of the value function of an optimal control problem. Although grid-based approaches provide a general-purpose means of numerically computing this value function, the curse-of-dimensionality often limits their use in all but simple applications. In this paper, inner-approximations of the backwards reachable set are characterised through the design of a feedback control law for feedback linearisable systems subject to input constraints. This structure is exploited to propose a selection of control gains via an optimisation problem. The computational cost and memory requirement of the proposed scheme are tied to the number of system states in a polynomial manner. Thus, the proposed approximation scheme is computationally tractable for feedback linearisable systems of relatively high dimension.