All versions of the Sharpe ratio that I've seen seem to assume that the risk-free rate is constant, and the standard deviation of the excess return in the denominator simplifies to the standard deviation of the portfolio return. Is it appropriate to include instead the standard deviation of the risk-free rate and the correlation of the portfolio return and risk-free rate when calculating the standard deviation of the excess return in the denominator?
Sure! Sharpe ratio must be defined as the return per unit risk on a zero-cost position. The notion you are referring to achieves this by assuming borrowing at a risk-free rate before investing, so refinancing risks should matter.
On a side note, the Sharpe ratio of any ForEx strategy would implicitly have the stuff you mention accounted for.