Just to be clear, you state that you made changes to the HT813, applied the settings and then restarted the gateway? By gateway, are you referring to the Grandstream Device or Ignition? Making changes to the HT813 does not typically require any restarts to the gateway(Ignition), but rather the Grandstream HT813 device needs to be restarted for the changes to take effect. (Just making sure)
In general, it seems as if the calls are more consistent when there is a country code with the number inside of Ignition. (I know the Grandstream device can be configured to insert that, but I have not set it up that way)
It sounds like your issues are more of a communication issue between Ignition and the HT813.
Have you gone through the entire guide via Support? HT813/HT503 Guide
It is important to make sure you have the Primary SIP Server on the FXO Port page setup correctly.
Also, did the HT813 ever work for you?