I want my App Engine project to be able to receive/handle traffic that comes in from a custom port (i.e. https://www.my-appengine-project.com:12999/some-endpoint)
I'm currently using a Google Cloud external HTTPS Load Balancer, but it seems that you can only configure it to receive traffic on port 443.
https://cloud.google.com/load-balancing/docs/forwarding-rule-concepts#port_specifications
What can I do to make this work? I need to receive traffic on a specific port.
You cannot implement custom port numbers with the Google Cloud HTTP/HTTPS Load Balancer. The supported backend port numbers are 80 and 8080 for HTTP and only 443 for HTTPS.
Reference: Forwarding Rules and Addresses
If you want to modify App Engine to listen on port 12999 then you cannot use domain names (only IP addresses).
In summary, you will need to switch to another service such as Compute Engine.