Phil,
One reasonable solution is add two additional parameters for the bottom and top (or bottom and span) of your logistic.:
Your function: K / (1 + ((K - x.0)/x.0) * exp(-r * x))
becomes alpha + beta*K / (1 + ((K - x.0)/x.0) * exp(-r * x)),
This logistic spans Y values from -alpha to -alpha + beta.
You may have trouble fitting it, because Yhat will be undefined if alpha is not large enough and I would be surprised if there were high collinearity between \hat\alpha and \hat\x.0. However, the choices of alpha and beta are no longer arbitrary.
Best wishes,
Philip Dixon