hotel card/program: spend least to get free room
What is your end game here? More details about exactly what you're trying to do would be useful.
If all you need is a single hotel night and don't care about the property at all, just search around and book the cheapest thing available, including nonchain hotels, basic service brands like Super8, 3rd party rates that don't earn points, etc.
Not worth playing the CC game just to get a single night at any random hotel IMO.