Take a surjective but nullhomotopic PL map $f:S^1\to S^1$, and let $K$ be the mapping telescope obtained by iterating $f$.  Then $K$ is contractible, locally compact, and finite-dimensional, but for any triangulation, the only contractible subcomplex (indeed, the only contractible closed subset) that contains the first copy of $S^1$ is $K$ itself.  Indeed, in order to be able to contract the first circle, you must take a subcomplex that contains the entire second circle, and to contract the second circle, you must contain the third circle, and so on.