Hi Brian,
You're right. This is a bit unique and something I have not been asked about before. I think the first place to look would be the JED for a download manager that does this already.
extensions.joomla.org/search?q=download+manager
It would take some time to go through them and make sure they do this.
You could always get a developer to write a custom plugin for you. This seems like a script that could be written since it's a common practice with all types of downloads.
One thing you could do for expediency is set the download link to open in it's own window and put a notice on the page with the download link that they can expect a new window to open and they should wait for it.
If I see a way to do this or someone else on the team is more familiar with doing it, we'll let you know.
If you find something, be sure to share it on the forum.
Cheers,
Ed