With the advice you gave me I got it working easily. On the single-channel version of the chip the SDI and SDO is multiplexed on one pin, but we only connected the SDI pin on the Arduino to it and left the SDO pin on the Arduino unconnected (according to the advice I got this is okay since we don't want to read anything from the chip anyway).
However to get the same response (steering and throttle response of the RC car) as with the analog potentiometer, we have to use a much wider resistance range with the digital pots than the range provided by the analog pot, which I found strange. Isn't the digital pot to supposed to work pretty much exactly the same as the analog once it is set up? As the transmitters pulse signal going through the pot is only 50 Hz, I don't think the frequency should cause problems.
Anyway, for now we managed to tune the response by trying out different values. So it works, but not exactly like I expected it to.