Transfer function of frequency shifted signal
How can I express this like \$L(s)=H(s)I(s)\$?


The first thing that jumps out at me is that \$l(t)\$ is not a linear or time invariant function of \$i(t)\$.

Now, \$H(s)\$ is the transform of the impulse response \$h(t)\$ which is just \$l(t)\$ when \$i(t) = \delta(t)\$:

\$h(t) = A+\dfrac{P}{CK}\delta(t)-\dfrac{P\delta(t)-CK(O-A)}{CK}e^{-Kt} = A(1 + e^{-Kt}) - Oe^{-Kt} \$

\$H(s) = A\dfrac{2s+ K}{s(s+K)} - O\dfrac{1}{s+K}\$

But, we can write \$L(s) = H(s)I(s)\$ only if \$l(t) = h(t) * i(t)\$ which is clearly not the case here.