I have been able to hack into many hotel TVs and use them with my FireStick via the HDMI. Look on the back of the TV for what looks like a phone connection. This will need to be removed since it controls the TV functions. If you are lucky they use the TV vendor's remote and after the phone jack wire is removed you can just use the remote to change inputs to the HDMI port you are using. If not, I carry with me a Samsung remote and a Universal remote. Most of the TVs I encounter in hotels use Samsung TVs, so that remote works. If not, I just have to look up the TV codes and program the Universal remote. Make sure you put everything back the way you found it when you leave of course.
I am currently in a Marriott TownePlace Suites and have done the above. The Samsung remote works here and I can watch whatever I want on my FireStick. An additional benefit is that when it rains, the tv signal always goes out. With my setup that is not an issue since I am getting content from the web.