Only seeing the wrong AirPlay device
I don't have an Apple TV in front of me to direct you to the exact place for this, but I'd trouble-shoot this like any networking discovery issue. Make sure your computer's subnet and the subnet of your Apple TV are the same. The hotel may have it configured strangely.

To get your computer's subnet go to System Preferences &gt; Network &gt; Wifi and click the Advanced button then go to the TCP/IP tab. Your Subtnet Mask is shown there and it should match the one the Apple TV is using.

Also (if possible) try and ensure your Apple TV and your Mac are both updated to the latest software available. Apple's support pages also recommend resetting the Apple TV, but I understand that may not be possible in a hotel.

Honestly your best bet is to contact the hotel front desk, since they will be able to do more advanced resets if needed and change the subnet as well.