Answer
Yes, achievable purely as configuration: right-click in the whitespace near the form fields you want to change and choose Properties, go to the Layout tab in the Properties window, check 'Use Customised Screen Layout,' then select or deselect which specific fields should show, optionally repositioning them on the page as well, and click OK to save. This can be done at an individual user's personal layout level for a one-off change, or, to roll the same layout out across the whole organization, by making the identical change while editing the Base Profile itself and saving it there instead.