Actually, our Express gateway should be leaving your orders as Processing. It wouldn't automatically update to Completed unless there is something else in place doing that.
That said, according to the highlighted lines in their code it looks like it should be updating the "mark_referral_complete" function any time an order is set to either Processing or Completed, so it should be handling both of those.
We'll have to install that plugin and play with it so see if we can figure out what's going on. It looks like they're simply using the "woocommerce_order_status_completed" and "woocommerce_order_status_processing" hooks that are built into WooCommerce directly. As I said, Express orders are setting the status to Processing so that should be triggering that hook just like anything else.
We're currently working on a few things to get 1.1.6.3.8 (which may just become 1.2) released, but I have added this to our GitHub repo so we can look into it after that.