Checkout Promotions supports 1-click payment with PayPal but it requires "Automatic Billing" to be enabled on your PayPal connection with Shopify. Automatic billing (also known as Automatic payments) on post purchase promotions allows Shopify to bill customers without requiring payment information to be re-entered by the customer.
You can check if you have Automatic payments enabled by going to your store's payment settings under Settings > Payment > PayPal Express and it should show Approved. If it shows as Pending, it may take 24-72 business hours from when you installed Checkout Promotions to when it gets approved. Check your email or reach out to PayPal if Automatic payments still shows as Pending after 72 hours: