To my knowledge, Bedrock is not compatible with Flywheel’s hosting offerings.
Since you’re hosting with Flywheel, you may want to reconsider your use of Bedrock and try a standard, non-Bedrock WordPress installation in Local.
Unless you have a specific project requirement to use Bedrock and are familiar with Composer, I highly recommend using a standard WordPress installation.
With that said…
Based on your screenshots, it looks like the Bedrock files are not inside the site’s app directory. Now that Composer is installed, you need to run composer create-project roots/bedrock while inside the site’s app directory.
In your previous screenshot where you ran composer create-project roots/bedrock, I can see that it was ran from ~ which is your home directory and not the site’s directory.
To run it from the site’s app directory, first right-click on the site in Local’s sidebar and then go to “Open Site Shell.” Doing so will open up Terminal and automatically change into the site’s app/public directory. However, since Bedrock needs to be installed in the app directory and not app/public, you will first need to go to the parent app directory by typing cd ../. Then, run composer create-project roots/bedrock.
Once you do that, follow the instructions under " Update .env" in the following article: Bedrock with Local | Bedrock Docs | Roots
If you’re on Lightning (Local 5 and newer), you’ll need to follow the steps I outlined in this post rather than the steps under " Update Local’s nginx config" in the article above.