Interesting. I got the exact same adapter (except the silkscreen is white). I trying recording two different HDD images (Time Crisis 4 and Cobra: The Arcade) on a 128GB card and attached it to my System 256. Nothing. The power and card detect LEDs light up, but not the activity LED.
I tested the card on an old PC (with another boot image, of course), and it worked fine there.
You asked about setting it as cable select. In theory, this should be detected as Master. You can force a cable select by cutting a pin off the cable for the *second* connector (it was pin 28 IIRC). But anyway, I don't know if it'd make a difference for System 256. The original DVD-ROM is physically set to Master via jumper, and works.
(edit: answering to myself -- it doesn't. I changed the jumper on the DVD-ROM to cable select and it still boots as Master. System 256 still loads from it just fine)
I researched this card ("SINTECHI") and found a nice article about it (
https://goughlui.com/2019/02/03/tested-generic-sintechi-fc1307a-based-sd-to-ide-adapter-sd35vc0/). The author came to the conclusion that there isn't much to do or modify it. He even dumped the firmware, but couldn't do much with it. I looked the chipset up, and the datasheet for the FC1307A chipset is virtually impossible to find. This chip is from around 2006-2007. All I could find was the datasheet for the earlier model of this chipset (FC1306T), which did the same as this one, but with older flash formats (like Smartmedia and Memory Stick). From what I could gather, there's absolutely nothing to configure on that one, it's all hard-coded in the firmware. And I guess FC1307A will be the same.
So unfortunately, this card does not work at all with System 256. What would make it work? No idea, as there doesn't seem to be any other form of debugging available for it (or is there?).