Ah. Per MSDN on NewFromFormTemplate:
This type or member can be accessed only from code running in forms opened in Microsoft Office InfoPath 2007.
So, we can't use that method in browser forms. Perhaps you should pass the URL of the form in and open a new browser window instead? I searched around a bit and found this:
System.Diagnostics.
Process.Start(http://server/library/Forms/template.xsn);
The form has to be set to full trust to run, but since you are having to admin deploy already, that may not be a deal breaker. There may be a better way to open a new form from a browser form, hopefully someone else will add to this thread.