slim4-skeleton VS dispatch

You can have the container call in a config folder with a file called dependencies or how ever you want. A few existing examples of this is here - Slim calls this app/dependencies in this skeleton project and Odan's Skeleton has this as config/container in their project.

Yes, as an example, the First Application's business logic (not that it's much) doesn't tie itself to the framework, as one example. Odan's Slim Skeleton, ADR Example (yes, I know you are the author), and the Aura Payload example class can be seen as other examples.

Even just the tests in your first link are very useful to see, such as: https://github.com/odan/slim4-skeleton/blob/master/tests/TestCase/Action/Customer/CustomerCreatorActionTest.php

Another option, to me and speaking to the OP as an alternative, it is simpler to add some Data Mappers/Repositories to abstract away the Database code. Good examples is the sample ADR application, Payload library example, Slim 3 First Application, or this Slim 4 Skeleton application

You could also use a Slim Skeleton like Odan's which incorporates much more features to help - docs.

Try Slim again. The setup may be too much, but the concept is the same. Look at the first application or any of the skeleton apps to help (https://github.com/odan/slim4-skeleton for one).

Dispatch - https://github.com/noodlehaus/dispatch

If OOP too much at this time, here is a procedural framework like Slim - https://github.com/noodlehaus/dispatch

Why though? If your customer is used to procedural code, you could still use classes, if you choose, in the base framework. Or is this an issue with keeping up with PHP 4 support per the readme? You've tested your "framework" with 5.3, so stick with 5.3 (go higher, please) and cut off PHP 4 support. Then, at least you can use Slim v2 to start. Still want procedural - here is noodlehaus/dispatch as a base.

You can also use this for a simpler API to help you get started- https://github.com/noodlehaus/dispatch