What is the make and model of the TV? Most likely your TV can receive what they call a true 4K resolution but displays it at 3840 x 2160 so the detected resolution would be accurate since it can accept the signal just not display it at the native resolution.
Adding a custom resolution you need to get the horizontal and vertical correct it would be 3840 x 2160 not the other way around. Using the base mode should work you shouldn't need to touch the detailed timings. You would need to pick the correct timing standard.
It's a 2015 Vizio M-series C2. Yes it can apparently receive the 4096 X 2160 signal (it will scale the image down). The 4096 limit is most likely incorrect because the reported refresh rate limit is also incorrect and my other devices indicate the limits given by the VIZIO.
I tried using the custom resolution feature, but that doesn't seem to be a solution. I eventually got it outputting the correct resolution by manually selecting it and then changing the monitor refresh rate. Simply changing the resolution setting isn't enough because the card would continue to output a 4096 X 2160 signal regardless (as reported by my TV), and the aspect ratio would be distorted. Changing the refresh rate seems to be a way to get it to reset and output the correct resolution. However this doesn't solve the EDID info problem.
At the moment, the only problem remaining is that the card output would occasionally turn to a solid black for no apparent reason. The computer would continue to function normally and I would still have sound, but the image would be frozen black The only way to fix the issue is to either reset the computer or graphics driver.