This is the right way to do it. Even better: because these settings are only necessary in the LMS, and not the CMS, you should use the “openedx-lms-common-settings” patch instead of “openedx-common-settings”. (but the latter will work nonetheless)
Values from lms.env.json are picked up by edx-platform/lms/envs/common.py. These values are then overloaded by tutor/production.py. The “openedx-lms-common-settings” patch (and “openedx-common-settings” as well) add content to tutor/production.py. So the latter solution should be working.