I'm not sure you fully understand the specifications you are quoting. The Apple TV supports L4 main profile, while L4 Main supports datarates of up to 20 Mbps, it does so with one eye on 1080p video at 60+ fps and 2K video at 30 fps. If you are producing movies for the Apple TV at 20 Mbps you really should be thinking of changing your export settings.
Nonetheless, it's not the Apple TV that is having a problem with your video but rather your network.
The following article(s) may help you.
Wifi Diagnostic Software (for Mac users)
You may also find some help on this page, where I’ve collected some of the more unusual solutions to network issues.
When making adjustments to your network for better optimisation, you may find some of the points mentioned on this page helpful.