Find transfer function from output
The impulse response \$h(t)\$ is the time derivative of the step response.  Thus, for the step response given

$$h(t) = \frac{d}{dt}S(t) = 4e^{-4t}u(t) + (1 - e^{-4t})\delta (t) = 4e^{-4t}u(t)$$

Then, the transfer function \$H(s)\$ is the transform of the impulse response which is as you alternately derived,

$$H(s) = \frac{4}{s + 4}$$

Finally we have

$$y(t) = h(t) \star x(t) = \int_0^t h(\tau)x(t - \tau) d\tau = \int_0^t x(\tau)h(t - \tau) d\tau$$

and

$$Y(s) = H(s)X(s) $$