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.
You can even visualize this example pretty easily. Take a cylinder, and deform it so that one end of it looks like a crescent, and then collapse the crescent to a circle by gluing together the two "C" shapes that make up the crescent and gluing the two endpoints of the "C". Glue that circle to one end of another cylinder, and then deform the other end of that cylinder similarly. Repeating this infinitely gives the telescope $K$.