I thought I should post an update, in response to my own question-in case anyone reads it and is still looking for an answer…
The issue was with my CMS and in-page dataset permissions. Once you add advanced permissions to the CMS table (including the right to “delete”), as well as read-write permission on the dataset where it is linked on the web page, the delete action then appears as an option for the button linked to the dataset.
I’m now working on trying to figure out how to add an “are you sure” pop up when a member clicks the delete button. This post has the answer in code, I now need to figure out how to get it to work on my page: Confirm "delete" when using the delete button? - #6 by admin-23