Hi Marcin,
Welcome to the data converters forum. We were just informed of your post.
Most likely, you are using either SPI-mode-0 (CPOL=0/CPHA=0) or SPI-mode-3(CPOL=1/CPHA=1) to read data.
To write to this converter, you need to use SPI-mode-1 (CPOL=0/CPHA=1) or SPI-mode-2 (CPOL=1/CPHA=0). In these modes, data can also be read, but the host needs to clock in data from the converter in hold time, which is easily done by an SPI peripheral automatically, but it becomes difficult with GPIO pins programmed to mimic SPI.
Regards,
Rafael