I believe you need to take the Airpods out of the case and leave them outside the case (unless of course the battery runs out, then you can place them in it temporarily for charging), the tracking doesn't work when they are stored in the case AFAIK. (But the location was already transmitted, so they'll see this location regardless of whether the Airpods are in the case or not.)
a white window appears on the phone screen stating "Bob's AirPods Pro - Find My"
Does that allow any interaction at all? If it was marked lost, there would be a message if the owner set one. Correct, there is nothing you can initiate yourself other than keeping them connected outside the case and checking for a message temporarily. And if the exact location is transmitted you might get a visit from the owner.
You don't have to keep them connected all the time, the location was already transmitted, it's more so that you can periodically check if the owner finally marked the airpods lost and set up a message for you to read.
If nothing happens after a week then you could hand it in somewhere to a lost&found. If you do, bring the iPhone along and connect them one last time outside the case. Then the iPhone will send this final location, and if the owner ever ends up visiting this location, they should end up right at the front doorstep of the lost&found. You can do this immediately if you don't want to wait.
You might want to call Apple Support.
Unfortunately Apple cannot assist with tracking or recovery. This is 100% up to the owner to use the software tools provided by Apple to do this themselves. You can call Apple Support but they'll likely just recommend you hand it into a lost&found.
Sadly users often don't know they have to do this, so they might not even know they can leave a message. Perhaps they'll need a bit of time to ask their tech savvy friends what they should do, and hopefully then someone will show them how to mark them lost and leave a message.