See Page 2 of the updated file below for a tweak to your approach. First, I selected everything in the repeater cell and grouped it, then cut & pasted the interaction code you had on the background widget to the group. This means that hovering over either text widget, or the background, and most critically the gear shape widget, will show the gear. As you had it set up, the gear is shown when cursor is over the background and hidden when cursor leaves the background–but when the cursor is moved over the gear it is no longer (detected to be directly) over the background, so the gear is hidden before it can be clicked.
Also, see Page 3 for a different approach, which uses a separate dropdown widget, moving and showing it as needed. This works if the dropdown menu is always the same regardless of the repeater content, and you don’t want the repeater cell to change size (pushing lower rows down to show the dropdown.
repeaterdropdown.rp (81.7 KB)