if I have a sinusoidal current as the input
You don't normally have a current driven primary, you have a voltage driven primary and, as saturation starts to kick-in, you get a higher current than for an unsaturating primary. The two effects of increased current at the peaks and decreasing dB/dt (at the peaks) I guess will tend to resolve themselves to something like linear volts out. Here's a pretty picture from the web that supports this general idea: -
Note that current is not very sinusoidal yet flux, despite the non-linearity is seemingly sinusoidal. Here's what that site says: -
When a ferromagnetic material approaches magnetic flux saturation, disproportionately greater levels of magnetic field force (mmf) are required to deliver equal increases in magnetic field flux (Φ). Because mmf is proportional to current through the magnetizing coil (mmf = NI, where “N” is the number of turns of wire in the coil and “I” is the current through it), the large increases of mmf required to supply the needed increases in flux results in large increases in coil current. Thus, coil current increases dramatically at the peaks in order to maintain a flux waveform that isn't distorted, accounting for the bell-shaped half-cycles of the current waveform in the above plot.
If you looked at the equivalent circuit of a transformer you'd see that any distortion in the magnetization current (providing input leakage components are relatively small and the input voltage waveform can be sustained in shape) won't affect the voltage across the ideal transformer: -
I\$_M\$ is the magnetization current and this current doesn't contribute to the creation of V\$_S\$ even if it is really badly shaped and peaky. Providing Rp and Xp don't generate too much of a volt drop (that would reflect the horrible shape of Im) then I think distortion will not be as bad as you believe it might be.