Hi Jacob,
There is nothing wrong with mapping, it just has its own limitations:
1) Data Mapping will populate the mapped fields automatically if the child record is created from the parent form.
2) However, additional logic (Workflow, other) will be needed in order to populate the child’s fields if the child record is created directly as a new record.
Here are the steps to create a workflow.
1) Go Settings > process to create one real-time workflow on the Quote entity:

2)Trigger--creation and fields change

3)Condition--check if the client lookup field is filled

4)Update record:


5) Save and Active.
