Hi Kevin
I've discussed this scenario with internal developers. They've confirmed that the LTI Launch won't send the start/end date that you've sourced via the API route you specified. As you identified, the custom parameters only send static info, so can't be leveraged to dynamically send the Start/End date.
One potential workaround - depending on the scope of the start/end dates - is to configure an External Learning Tool for that course specifically. This goes against the typical practice of setting up an External Learning Tool at the Org level and sharing it with child courses. But this configuration gains you the ability to specify a static start and end date for the course, which could be sent as a custom parameter on the link. Again, if the start/end dates are topic-specific (rather than course-specific), this solution likely won't be granular enough.
Let me know if that's approaching an appropriate workaround. Or if I've missed the mark entirely, please provide any additional details that can help flesh out my understanding of the scenario so that I can look into alternative approaches.
~Sarah-Beth