Sounds to me like a session timeout issue, which can occur if traffic is flowing only into one direction using SSG. The returning traffic is let into the LAN immediately, I suppose, instead of passing SSG again. Either the Cisco has to transfer all traffic for local to SSG, and SSG is default gateway; or Cisco is default gateway, but passing all non-related traffic to SSG, which does Internet.
IMHO best config would be SSG as default gateway, Cisco for MPLS connected to an own interface on SSG.