Have you looked at the deformation theorem for rectifiable currents? This essentially states that any integral current $S$ can be approximated by a polyhedral current situated not very far from $S$. Your smooth chain defines an integral current. A good place to look for more details is Frank Morgan's Geometric Measure theory. A Beginner's Guide, Section 5.1. I believe that the strategy used in the proof of the deformation theorem could be useful for your problem too.

Have you looked at the deformation theorem for rectifiable currents? This essentially states that any integral current $S$ can be approximated by a polyhedral current situated not very far from $S$. Your smooth chain defines an integral current. A good place to look for more details is Frank Morgan's Geometric Measure theory. A Beginner's Guide, Section 5.1. I believe that the strategy used in the proof of the deformation theorem could be useful for your problem too, or at least the weaker version suggested by Matthias Kreck.

More precisely, the deformation theorem indicates that your chain $c$ can be approximated (in various norms on the space of currents) by a nice polyhedral chain $c'$, whose support can be chosen in an arbitrarily small neighborhood of the support of $c$. In particular $c'$ is homologous to $c$, if $c$ is closed.