Unfortunately it's not possible to add the payment method of a order to the default packing slip template from Shopify because the payment gateway variable is not exposed to the template.
You can see the list of the liquid objects available on the template here https://help.shopify.com/en/manual/orders/packing-slips-variable-list
You might be able to add it if you use an app to create the packing slip, for example Shopify's own Order Printer app: https://apps.shopify.com/order-printer
Let me know if this helps you!
Hey guys! I had the same problem and found a great solution with the Order Printer Pro app.
I highly recommend it, they helped me customize the code so the payment method shows up right below the date. Hope this helps! vanessa