Picard is not really meant to write only a single tag, but if cover art is really the only tag you want to write you can do the following
If you are Ok with Picard writing a tag if it does not exist yet, but you don’t want to overwrite it, then you can add that tag to the preserved tags in Options > Tags
If you absolutely do not want a tag to be written at all, add a script
$unset(tagname) in Options > Scripting
I’d recommend the first approach for tags like artist, album, title and tracknumber, as those are used in the UI and inserting them makes Picard hard to use. Also your existing files likely already have those tags, and if not they probably should have them.
You’ll need to list all tags you don’t want Picard to write.
And I still would recommend you let Picard write the MusicBrainz IDs to the tags, at least the recording and album ID. That way you can e.g. easily update the artwork again later without needing to match the files again.
Please also search the forums here, there are a could of threads about configuring Picard to only write a few tags.