As you have currently got it set to "Authorize" all that does is check with the issuer if you are going to have permission to capture the payment in future.
In order to have the Invoice generated you need to set the ApplePay "Payment Action" to "Intent Sale" (Capture). Doing this will automatically generate an Invoice for you and if you look under "Transactions" for the order it will change from Type=Authorization with order status "Processing" to Type=Capture with order status "Complete" (or whatever you have configured it to be).
Note: Depending on your business requirements you might want your ERP system to create the invoice, or maybe even invoice-on-shipment. If this does not apply to you then the above solution should probably be fine for you.
You can manually invoice from the order view in the admin by clicking "Invoice" then on the next page "Submit Invoice" ("Capture Online" should be selected, this is the default)
In terms of orders not syncing to the ERP system it really depends what causes orders to sync. I have worked with ERP systems that use a variety of methods to sync data with Magento, so you should check what triggers the sync e.g. external system connecting to the Magento API, or a module installed on your site. Additionally you should check the criteria for synchronised orders e.g. invoiced, order status, order state, etc.