Do SDIO bus speeds need to match between two devices?
as you said, the CPU handles 200MB/s at a clock speed of 200MHz, that is assume a 8-bits bus clocked by 200MHz. the SDIO is 4 bits data bus, therefore at 208MHz SDR, it gives 104MB/s.
in order for it to work, you will need to interface 4bits data bus from CPU to SDIO at a clock speed it can accept. It is OK your CPU runs interface clock speed at 200MHz if that is the MAX it can provide. but since SDIO is only 4 bits bus, so your CPU side will have to follow that throughput.
usually uC have SDIO dedicated interface, that should interface with standard SDIO device.