The last part of your reference is MLS_Name_r.Id. I think you want MLS_Name_r.Name

I tried that and it does put the correct name in field but when I try to save it says

Ohhh… I see what you are doing now… try two actions in the action sequence. The first action updates MLS_Name_r.Id and the second action updates the same field to MLS_Name_r.Name. I think that will work based on a use case I had a long time ago so I may not be remembering it correctly.

Same results.

Here is what works  {{{$Model.TempEngagement.data.0.Temp_Engagement_Location__r.Zip_Postal__r.MLS_Name__c}}}

Leave off the Id and use __c instead of __r