FireTV 3rd-gen Pendant is probably your best choice. FireOS has a captive window portal browser that comes up when you join a wifi network, which will let you login to the hotel wifi with your room details. Then the Channels app can be used to connect to your home DVR.
Note that some hotel wifi does not allow peer-to-peer communication (i.e. connecting from your phone to the firetv for the login), so it would be best to do the “Login to remote dvr” at home first and then you can use “reconnect to remote dvr” from the hotel room.