How can shot noise be dependent on dc current, not instantaneous average current?
Shot noise is defined by the magnitude of the average value of all current sources, both static and dynamic. You can find the average value of the shot noise current by adding the static current with the average of each of the individual current sinusoids. The total current that may exist is (assuming dynamic sources of current are sinusoids): $$I_0 + I_1cos(\omega_1t) + I_2cos(\omega_2t) + ...+ I_ncos(\omega_nt)$$

Where n is the largest sinusoid you're willing to look at (ideally infinite).

That being said, the current magnitude you're looking for is

$$|I_d| = |I_0 + \frac1T_1\int_{-\frac {T_1}2}^\frac {T_1}2I_1cos(\omega_1t)\ dt + \frac1T_2\int_{-\frac {T_2}2}^\frac {T_2}2I_2cos(\omega_1t)\ dt +...+\frac1T_n\int_{-\frac {T_n}2}^\frac{T_n}2I_ncos(\omega_nt) \ dt| \text{,   where} \ T_1, T_2,...T_n \text{ is the period for sinusoid n}$$

More simply,

$$|I_d| = |I_0 + I_{1_{average}} + I_{2_{average}} +...+ I_{n_{average}}  |$$

And finally the classical shot noise is,

$$S(\omega) = 2q|I_d|$$

Note that the sinusoids can be noise components, but we can still find the average, or average DC value of the sinusoids, which we did.