So it would be
convenient if pending members could be "claimed," just as editing off a
pending message is claimed, to prevent other mods from trying to go into their
page and set it up.

In my group we have a "workshift", i.e. each moderator has an assigned day (occasionally we have 2 moderators on the same day). If a moderator for any reason wants/needs to follow a specific pending member, he/she leaves a note in the Notes section of that member's page.
Before handling pending members (especially those who have been on the pending list for some days), moderators check their Notes section.
I don't know if this would work with your group. Hope this helps, anyway.
Best,
Marina