I use LocalWP with Bedrock configuration all the time. Here are my tips:
Move all of the contents of your bedrock project directly into the app folder. You can leave the /public folder there, but delete all of its contents (this just makes it play nice with the local site shell as it will look for this directory). If you’re using git, add /public and /sql to your .gitignore file.
Then, set your conf/nginx/site.conf.hbs file to:
root "/Users/myusername/Local Sites/bedrock/app/web/"
Your Database tab will look like it does in your screenshot, but it will function fine. If it really bothers you, you can add a dummy wp-config.php file with those defaults in the /public folder. This serves no actual function for WordPress, but it allows Local WP to detect it.
If you get a warning that the Site URL / Home URL don’t match, I usually just manually change them in the database using Adminer because WP will use the .env file for those variables anyways. Set them both to your URL (ie. https://bedrock.local) without the /wp for the Site URL.
Hope this helps!