A quick rough draft that I just got working in-house:
tell application "Curio"
external hyperlink as markdown of selected figure of active document
end tell
Selecting a figure in Curio and running that script returns this string:
[Curio project \"Pepsi Campaign\" > \"2020 Plans\"](curio://$Projects/Pepsi%20Campaign.curio?i=hVStpJFZRyOG1-qc5bIs_Q&f=iUuzRZ7dQ3OeqrXja5flug)
Thus the link name is:
Curio project "Pepsi Campaign" > "2020 Plans"
And the direct URL to that figure:
curio://$Projects/Pepsi%20Campaign.curio?i=hVStpJFZRyOG1-qc5bIs_Q&f=iUuzRZ7dQ3OeqrXja5flug
Can you work with something like that?