There isn't a way to force the URL to display the Unique ID rather than the API ID, for the API ID is the only immutable ID that Jama items carry (Unique IDs can be changed when you alter a set key, for instance, and Global IDs depend on an item's sync status, if any). You could use /abstractitems to get items with a specific documentKey, e.g.
response = requests.get('http://localhost:8080/contour/rest/latest/abstractitems?documentKey=AGILE_-STY-4', auth=('sample', 'password'))
(In this case, sample and password are the user and the URL is a placeholder for a local instance). However, I don't know how that would help you achieve your end goal, but it's really the only bit of information I can offer. We unfortunately don't have resources for using Jama with many development tools, only Subversion. Hopefully someone subscribing to this section has overcome the same issue and can comment.
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