I was finally able to get this working by setting the registry key "HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\WindowsNT\CurrentVersion\Windows\USERPostMessageLimit" to 1,000,000 from the default value of 10,000. The error "Not enough quota was available to process this command" is triggered when this PostMessage limit is reached. PostMessage is an underlying Windows API function that places messages into a queue. It can be called from such things as Document.OpenAndActivateDocument. The limit is for how many PostMessage calls can be waiting in the queue at once. Since this is being run up by the underlying Revit API code for DWF Exporting and document opening my only hope was to increase the limit.
However, I do recommend that you try this on a single machine first, as I am doing, because I'm not sure what kind of other side effects it may cause. If you have the time then test reducing the amount and see if it still works on your biggest set of models and then stop once you've found the sweet spot. This may take a while though because each time the registry value is changed you will need to restart your machine.