WhatsApp can work independently from Google services and maintains a local backup of any conversations you make.
The following should work:
Install the app, but don't open it. Copy the "com.whatsapp" folder from the "Android/media" folder from the device you want to transfer to the "Android/media" folder on the device you want to transfer to. Allow the app access to the "com.whatsapp" folder by changing its permissions in the settings (just allowing it access to this specific folder via storage scopes should work, but full access will work also.)
You can now open the app and walk through the setup. The app should detect the backup.
Unsure if the storage-scopes-before-opening approach is needed 100%, but other users (including myself) have reported difficulties with regards WhatsApp detecting such backups otherwise. I will test further and report back if this has changed.