I tried to display In-App Messaging but it didn't show up with TWA. In-App Messaging works without any problems with normal Activity.
I use https://github.com/GoogleChrome/android-browser-helper/tree/main/demos/twa-basic to test TWA.
My application is correctly configured with Firebase.
I created a campaign.
My logs after publishing the campaign:

I closed my application and then launched it and I didn't see In-App Messaging.
I tested another application with a standard Activity and there was no problem displaying In-App Messaging.

I took a lot of time today to answer this question for myself.
In my opinion TWA can't work with In-App Messaging. I am not an Android programmer and I could be wrong.
This is my test Activity:
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To display In App Messaging we need a Activity. I think we can only In App Messaging display on splash screen TWA but i didn't try it.