Well, it seems, somehow if we pick "Uninstall.exe", the Launcher can identify the version from there. So this is how I treated Wwise Launcher to do the job:
1. Locate "Unity.exe" in the file browser. For example C:\Program Files\Unity\Hub\Editor\2021.3.1f1c1\Editor
2. Rename "Unity.exe" to "Unity2.exe"
3. Rename "Uninstall.exe" to "Unity.exe"
4. In the launcher, pick the fake "Unity.exe". The launcher should identify the version now (at least for me).
5. Click "Integrate".
6. Act quickly!! While the launcher is "Validating Project", rename "Unity.exe" back to "Uninstall.exe", and "Unity2.exe" back to "Unity.exe". You must finish this before the launcher reaches the step of "Launching Unity to finish the integration".
And it works for me!!
(Well, Audiokinetic, please fix the issue. I submitted a ticket. )