I was about to say it looked like 3-wire SPI might work, but after going through some of the Philips stuff, I decided that might be too simple to work without some bit banging.
Table 1 seems pretty simple. That is, a 3-bit address (A0..A2) + data bits, which is not unlike 3-wire SPI to LCD's that aren't being read.* However, address b'010' (AGC gain) is expecting 9-bits (D0..D8) for data. Clamp reference, b'100', expects 10-bit data (D0..D9).
That said, I would still try sending 16-bits and see what happens (filling with leading or trailing zeros). You could bit-bang that to only 13 bits, but it may not be necessary.
Further digging at Philips might give some some example code, if not for your specific chip but for other Philips serial chips.
*The LCD in my current project accepts either 8-bit or 16-bit serial, depending on address. Except, the 16-bit is sent as two, 8-bit bytes. Not a big difference as you can leave CS low the entire time.