Thanks for your detail description, I misunderstand the requirement before, because we hadn't met the scenario like you mentioned.
In our routing we don't have existing APIs for it, but you can implement that by our product. After quickly research, it looks this should be a professional service task, please contact your sales for detail about that.
If you want to implement that yourself, as below is some thinking.
1. Calculate time for each segment by length and speed limit on it.
2. From your start time and the time in "Stop 8", we can know the target travel time before we arrive "Stop 8".
3. Create an algorithm to get all possible routing which takes time less than the target travel time when it arrive "Stop 8".
4. Optimize the result and get the routing you want.
Wish that's helpful.