Upon drop down, context menu like the following could appear with relevant options.
Link page <file name>
Link section: <# section heading>
Link block: Lorem ... laborum.
Link block: consectetur ... elit
“Link page” option would be directly under cursor unless it is not possible due to drop position being near bottom edge of screen. (or top edge when the list is implemented/displayed in reverse order.)
Hover over each option would display pop-up preview of corresponding link target to help verify correctness of choice, just like when hovering over normal links in markdown file.
To preserve current fast way of linking the page without need of additional click, I would suggest to display the menu only e.g. when dragging with some modifier key pressed or when dragging by right button. Developers, please select method that fits best with other UI.