Why can't I build AAB files with the development client?

As per the docs, when you set developmentClient to true, it creates a debug build and an artifact (apk) you can share internally, which implies you will need to set it to false in your case.

What’s the correct approach to testing in-app payments?

The right approach is to publish a build on the internal testing tack or the closed testing track first.

Summing up:

• Create a release build
• Create a new application on Google Play Console which will be the dev version of your app (with a different package name)
• Publish the dev app on one of the testing tracks
• Integrate IAP using a provider of your choice (eg. RevenueCat)
• Now run a debug build locally with the same package you published to the testing tracks
• If the purchases work as expected, repeat the above steps for the production version of your app

This answer has a helpful checklist you can follow along while setting up IAP: https://stackoverflow.com/a/76877489/14431966