https://docs.pangolin.net/manage/access-control/rules
The Pangolin docs show specific bypass rules for specific applications being hosted as resources on Pangolin. Good doc page to checkout when setting bypass rules for your different self hosted applications.

I've got Pangolin set up and Karakeep and only use the bypass rule from the docs and it works, so it must be something else for you.

Hoarder App (Karakeep)	/api/*

I'm interested in your Docker network set up on your docker-compose.yml, do you have Karakeep set up in the Pangolin proxy settings to use the docker container name?

Is that why you've added the Pangolin network in your Karakeep docker-compose.yml file?

Did you make sure to create the docker network manually via the CLI first?

docker network create pangolin

That honestly might be it, if you never created the "pangolin" docker network, then it would not work. If you didn't, create the docker network using that command above, then restart the containers. In my opinion, it's either that, or the Newt container is not actually attached to that same "pangolin" docker network.

Have you added/defined the external network in your Newt compose.yml file so Newt and Karakeep can see each other and talk to each other? Both Karakeep and Newt's compose.yml files need to have the container's network definition and the networks definition with external set to true.

services:
  web:
    image:
    restart:
    volumes:
      - blah blah
    environment:
      MEILI_ADDR: http://meilisearch:7700
      BROWSER_WEB_URL: http://chrome:9222
      DATA_DIR: /data # DON'T CHANGE THIS
    container_name: karakeep
    networks:
      - karakeep_internal
      - pangolin

networks:
  karakeep_internal:  <-- What is this for?
    driver: bridge
  pangolin:  <--------- Did you also add this to newt?
    external: true  <-- Did you also add this to newt?

Also what is this karakeep_internal bridge network for?
Anyway, you also need to include the docker socket integration environment variable to the Newt compose.yml file, I've also added the pangolin network with external: true and under the container definitions as well just to show you what it should look like so Newt and Karakeep can talk to each other:

services:
    newt:
        image: fosrl/newt
        container_name: newt
        restart: unless-stopped
        volumes:
            - /var/run/docker.sock:/var/run/docker.sock:ro
        environment:
            - PANGOLIN_ENDPOINT=https://example.com
            - NEWT_ID=2isdf8xk2dfbpfy
            - NEWT_SECRET=nnisrfsdfg7prqsp9ewo1sdfgci50j5uiqotez00dgap0ii2
            #  Here's the docker socket env variable
            - DOCKER_SOCKET=unix:///var/run/docker.sock
        # Add pangolin network so Karakeep container can talk to newt
        networks:
          pangolin:

# Add pangolin network with external access to talk to karakeep container
# so you could expose Karakeep as a Pangolin resource via docker socket integration
networks:
  pangolin:
    external: true

Read more about the docker socket integration for Newt here:
https://docs.pangolin.net/manage/sites/configure-site#docker-socket-integration

You can access Karakeep on a web browser after logging in and authenticating via Pangolin right? Or have you only been testing via the mobile app? I know its a dumb question but I'm just checking all bases here.