That is why I got you to do the test.. I want to eliminate one or other of the boxes to prove where the issue is.
The only thing left is permissions. But it seems that Apple TV has no issues with the main movie file.. so it does not kinda make sense for the permissions to be a problem.
Nevertheless I would like you to fix the permissions. The easiest way to do this is via old airport utility, 5.6.1 on suitable Mac or PC.
I guess your first problem is to run 5.6.1 airport utility.. you need older Mac OS.. early Lion.. or special script from internet I cannot post URL but just google for airport utility 5.6.1 launcher.
Any windows computer you can use.. or bootcamp partition on a Mac.
If you go to the Disks tab, File Sharing.. by default Airport Disk Guest Access is Not Allowed.. change it to Read and Write.. and click update. This may have multiple affects I cannot predict. So if you have other files that should not be available don't do it.

Unfortunately I don't have recent Apple TV to check and reproduce your problem.. but my guess is the TC is just unsuitable for this job.. let me check what alternatives we have.