For those it may help, I got about 85% of the way by creating a Zapier rule. The zap kicks off when a new task is created, or modified, under a certain project in the client account. When initiated it searches to see if I have the task already, if not it creates a new task in my instance of Asana including the originating task body information and due date flag. If I already had that task it updates to reflect the changes made on the client side.
The last 15% I haven’t nailed, I can’t get attachments to port into my instance and unfortunately for me porting over attachments is a must. Thought I’d share in case this Zap, without attachment capability, is still useful to someone else.