It seems to be an oversight in LR -- the metadata field "lastEditTime" is available to plugins via the Lightroom SDK, but you can't use that field to define a custom tagset for displaying in the right-hand-side Metadata panel.
However, here's a workaround that might work for you: In the right-hand Metadata panel, select the "EXIF and IPTC" preset, and the field Metadata Date will show the date/time of the last time LR wrote metadata back to the file. In Catalog Settings > Metadata, make sure "Automatically write changes into XMP" is checked. This will cause LR to write develop settings or metadata changes (keywords, captions, ratings, etc.) back to the file soon after you make the changes ("soon after" is typically less than a minute).
