From what it looks like, the join won’t be straightforward without the key, given that there can’t be much done on how the REST API returns the data. Is there any other way Qlik can extract these responses to avoid this issue?
If you are okay not leveraging the REST API and cool playing with raw event data, the long shot would be setting up recurring exports of your project data to the cloud destination of your choice ( S3, Snowflake, GCS ) directly from the Amplitude. Maybe then try to connect Qlik to these destinations and get the desired data using some basic SQL for DAU calculation.