I know this post is ancient, but are there limitations to what kinds of fields will show up in the Configure Allow List? I have two date fields related to image acquisition and I can't get them to even show up as checkbox options in the Configure Allow List. They are in the mosaic Footprint attribute table, and I have another field I added there (Text) that shows up fine. I see other services have date fields in their imagery service, so I'm stumped.
EDIT: Moments after posting, I figured it out. Before publishing, you need to first go in through Catalog in ArcGIS Pro and right click the mosaic, then go to Properties. Choose the Defaults menu from the left side of the window, and then set the Allowed Fields option to include your new field(s). Then, do the same thing when publishing (or overwriting) the service as described in the accepted solution above (Configuration -> Imagery -> Allowed Fields in the panel when publishing/overwriting).
EDIT EDIT: Oh lord, plot twist—I posted too soon! The fields were available in the Configure Allowed Fields for the MD properties in Pro, but when I went to publish, they were not available as options in the Allowed Fields section, there. I have no idea what the issue is.
-Aaron