As you suggest, in the case of non-stationary time series, the Hurst exponent is not suitable to measure the time seires persistence for the reasons you cited in the question. Particularly, when $H(q)$ is a non-linear function of q, as in the non-stationary time-series case, the time-series has to be analysed as it is a multi-fractal system (to deal this topic with, look at the 2nd reference below).
As regards the measure of the time series persistence for non-linear dynamics, I suggest to implement a detrended fluctuation analysis, that produce an indicator, highly correlated with the Hurst exponent, that is more appropriate to model non stationary, changing with time, data.
For more references about the topic, look at the Peng's seminal paper:
Peng, C.K. et al. (1994). "Mosaic organization of DNA nucleotides".
Phys. Rev. E 49: 1685–1689.
an to the Mandelbrot's book about using fractal distributions in the financial markets:
Mandelbrot, Benoît B., The (Mis)Behavior of Markets, A Fractal View of
Risk, Ruin and Reward (Basic Books, 2004), pp. 186-195