After a day of thinking of my architecture, I'm not sure reading anything will be possible. Especially if each app will be containerized. So my idea won't work.
The point is that I'm trying to migrate parts of the old application to the new, and adding this wrapper will increase security because I'd replace the old htaccess + plaintext passwords with Laravel login (which has hashed passwords in the DB).
So my new idea is this. Since I already have implemented Laravel passport into my app, I can use personal user tokens (which is similar to JWT approach).
When going to the old app, I'd need to check for this token in the request. If there is no such token I can redirect to the new app.
Then, when I log in to the new app (Laravel) I can check the referrer and after a successful login, I can redirect back to the old application with the generated token.
The old application will now have this token which can be used to ping the Laravel API to get the user details. If the call to the Laravel passes and I get the response with needed data, I can then proceed to set user session on the old app as the specified user.
Old app has users and permissions set up so I'll have to make some changes there so that the functionality works.
I keep thinking of ways to simplify this process, but the old app is in really bad shape, and I need this to work so that the client sees that I'm not just going: switch to Laravel because it's awesome, but because it offers tons of other improvements that current app doesn't have...