A Sqlite database is only one file, which you surely have access to.
So just add a Database configuration to your IDE and point it to the Sqlite file.
Click on the Database flyout on the left side then on the + (Plus) and add a Sqlite configuration.
(IntelliJ has good documentation on Sqlite Database connection as well of course)

Then you are ready to go.
The UI of the database structure is quite good.
You can browse the tables by doublelick on the table in the database structure view.
Also you can edit values in the table view.
Note: When the DB was created inside the Docker container you probably dont own the DB from the host and you wont be able write to it.
While developing just change permissions on the host, if necessary.

You can also open a Database Console window where you can test queries.
This is particular better than using Adminer because these test queries are persisted and will still be there the next day.
You can add multiple database consoles and reopen them from “Scratches and Consoles” in your Project menu.

So no Adminer needed. Sqlite is just a file, dude.
No hassles with docker-compose anymore for DB administration and still using Docker for the app. :)

Update 09.08.2023:
JetBrains has now also uploaded a nice Youtube Video Tutorial on how to deal with Sqlite databases in IntelliJ IDEs like f.e. PHPStorm:
Working with SQLite Databases in any JetBrains IDE

I learned that you can now Drag and Drop Sqlite database files into the DataBase Pane and the IDE connects it directly, nice!

Update: Today i would rather use the IntelliJ Database Tool for Sqlite administration.

But since Adminer is a popular hacking target they introduced a feature that does not allow to run adminer without a password, out of the box.
Sqlite database usually runs without password and dang, workaround needed!

So we need to add a plugin to adminer that allows password-less login and extend the official Adminer Docker image to include the plugin.

We add a script that loads the official password-less-login plugin and copy it to the plugins-enabled folder of the Adminer image

login-password-less.php:

<?php
require_once('plugins/login-password-less.php');

/** Set allowed password
 * @param string result of password_hash
 */
return new AdminerLoginPasswordLess(
    $password_hash = password_hash("admin", PASSWORD_DEFAULT)
);

Dockerfile:

FROM adminer
USER root
COPY login-password-less.php /var/www/html/plugins-enabled/login-password-less.php
#USER adminer # we run as root because of permissions problems on db file with the volume

In docker-compose.yml:

version: "3"
services:
  app:
    build: ./php
      - mailcatcher
    volumes:
      - "../:/app:rw"
      - "./php/cli/php.ini:/etc/php/7.3/cli/php.ini:ro"
      - "./php/fpm/php.ini:/etc/php/7.3/fpm/php.ini:ro"

  nginx:
    build: ./nginx
    depends_on:
      - app
    command: /bin/sh -c "nginx -g 'daemon off;'"
    volumes:
      - "..:/app:rw"
      - ./nginx/default.conf:/etc/nginx/conf.d/default.conf:ro
    ports:
      - '8088:80'

  adminer:
    build: ./adminer
    restart: unless-stopped
    volumes:
      - "..:/app:rw"
    ports:
      - 8080:8080

File tree:

• adminer
  – Dockerfile
  – login-password-less.php
• nginx
  …
• php
  …
• docker-compose.yml

Note that since docker-compose.yml version 3 the volumes_from directive was removed in favor of top level volumes to share volumes across images. However this implementation is imho rather weird, so i jut duplicated the volumes declaration in the App container and Adminer container.
Feels rather wrong but thats all i could come up with. Any advice here is welcome.

Now that this is done we can open the Sqlite3 database with the pseudo-password “admin” and manage the database.

After all you could also just copy the one-file Adminer with plugin to the php container and run it from within the php container. Just adjust your Nginx config to allow calling the file directly and take care to not deploy Adminer to live.
There is also a project that bundles the plugin with Adminer to one file: https://github.com/FrancoisCapon/LoginToASqlite3DatabaseWithoutCredentialsWithAdminer/blob/master/build-adminer-4-sqlite3-into-one-file.sh