I think the mode you want is Hotspot. This appears as one device to the hotel's system and allows you to connect multiple devices. You will use the wizard to associate to the hotel wireless. Then connect one of your devices to the 505. From there you should be able to open a browser and go to some non-https page, such as dlink.com, and the hotel's login screen will appear.
At least that's my theory - which I will be testing out this coming week, as I just received my DIR-505 yesterday. I have to say the manual is rather unclear on the difference between repeater and hotspot, and contradicts itself in various places. If one goes by the descriptions early on in the manual, repeater would be the choice, but I think in that mode it simply passes through the connections to the hotel's system, and each client would need to log in separately.