When exporting cases, however, this option does not seem to exist, so you'd end up having to copy in question labels manually. Does anyone know if there's a way to automatically replace question IDs with question labels under the display field for case exports?
Good question, though I'm afraid the answer is no. In case exports, the column names default to the property names themselves. The property names names are defined on the Case Management tabs of the CommCareHQ App Builder. Though those case property names default to the question id, they can be named or renamed anything you'd like.
However, if you do opt to remane case properties to values that are more useful to you, I advise taking note of the changes made as it may make tying out old exports with new exports more difficult.