Like you already noticed, some fields cannot be specified for "Field Mappings" in the lookup feature.
Therefore, you need to use some JS to accomplish this. Here's an example flow of how it can be done:
1. Obtain record data using the lookup field
2. Place the record number of the obtained data in one of the fields
3. Set a field change event(*1) in the field mentioned in step 2.
4. Within the event, using the record number as a key, obtain the target record data by API (*2)
5. Place the value that you obtained in step 4 in the corresponding field
Kintone Help - Setting Lookup
*1 Kintone developer network - Record Create Event - Field Change Event
Kintone developer network - Record Edit Event - Field Change Event
*2 Kintone developer network - Get Record (GET)
I hope this helps