A document can only exist on a single task at any one time as far as I'm aware, but you could for sure leverage fusion to move documents from one task to the next as required.
To make sure I understand, you are trying to update one of a parameter's options? If so, how are you making your call? You shouldn't need to do any of this when making an update, because your update shouldn't be in the url.
It's one of the modules that you can create, link: https://experienceleague.adobe.com/docs/workfront/using/adobe-workfront-fusion/webhooks-in-fusion/instant-triggers-webhooks.html?lang=en When you set up your webhook, configure it to watch for your triggering field to be changed.
Can you post your API call? I just tested it with the task constraint set to ASAP, and with all 4 of the duration types, and my body was {
"duration": 15
} and that worked.