How can I process Shipping and Pickup Club Orders separately?

You can manually process club member's packages in Step 3 of your Club Package. If you are looking to process Shipping Club Members first, you can create a filter in Step 3 for Order Delivery Method: Ship. Once this is configured and the list is filtered, you can click Process and it will only process the Ship Club Packages. You can then do the same for Pickup orders.

Once a club member is successfully processed in a Club Package, they won't be processed again. Each club membership can only process one order per Club Package.