Celtics2021 said that DET could use their full MLE to take in Hauser, even though this is a trade not a FA. Google AI also says this:
Yes, under the current NBA Collective Bargaining Agreement (CBA), teams can use their Non-Taxpayer Mid-Level Exception (NT-MLE) and Room Mid-Level Exception (R-MLE) to acquire players via trade. This allows the exceptions to function essentially like traded player exceptions.
This would hard cap DET at the 1st apron. Spotrac shows DET as being well below the 1st apron so it should work but it doesn't seem to on the trade machine. I believe that a straight trade of Hauser for #21 should work. DET still has to pay some players, but they would have a lot of financial flexibility still.