@EriksCat  Hello, 

From your description Hotel Wi-Fi is a Portal wifi (authentication page pops when connected to WiFi), as public/ portal Wi-Fi has different ways for authentication, range extender may fail to that Portal Wi-Fi sometimes. Here are some suggestions you could try since you already had the extender.
1.  Please confirm end-devices can pass the portal WiFi when directly connected to the Portal WiFi. If possible, please send the screenshot of the authentication portal.
2.  Check if the portal wifi only provides One voucher(code or username/password) and only allows one device to access the internet at the same time.
If it only allows one device, sorry to tell you RE could not help increase the number of clients,  it is recommended to connect the end-devices directly to portal wifi without RE. 
3. After setting up RE, if the end-device connects to RE by WIFI, does the Portal pop up? If you put in authentication information, is there any Error message? Please send us the screenshot of the portal authentication page and Error message.
4. Confirm RE is the on the latest firmware:  https://www.tp-link.com/support/download/re305/v3/

Do you get the chance to know the model of hotel Wi-Fi? Or a page of the Portal page set up? Thank you.