Adding phase to a signal that has multiple frequencies
Caveat: many behavioral models of amplifiers exist, none of which is perfect, and I don't know what is best for your particular amplifier.  But the most common model is basically your option 2.  I have never heard of anyone using option 1.

Let $x(t)$ be the input to the amplifier in complex baseband form.  A very common amplifier model is to assume that the gain is a memoryless function $g(|x(t)|)$ of the input amplitude, and the phase shift is also a memoryless function $\phi(|x(t)|)$ of the input amplitude.  The complex baseband output is therefore

$y(t) = g(|x(t)|)\exp(j\phi(|x(t)|))x(t) $

This model assumes the signal is relatively narrowband.

There is no need to decompose the signal into its Fourier coefficients to perform this transformation, and indeed this only complicates the matter.