mean(abs(pred - true)) / mean(abs(true))
I defined above metric that works well for measuring reconstruction error for zero-mean signal data (1D); does it have a name? It's not MAPE.
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It's very closely related to MAD/Mean, where you just take the mean of the series for the denominator, without taking absolute values. If your series is nonnegative, the two notions coincide.
MAD/Mean is a weighted MAPE, with the actuals as the weighting (Kolassa & Schütz, 2007, Foresight). Your MAD/MAV analogously is a weighted MAPE with the absolute values of the actuals as the weighting.
Whether or not this KPI "works well"... well, if your series is stationary, then it is equivalent to the plain MAD, which is minimized by the median. So minimizing this KPI will incentivize you to report the predictive median. If your series is symmetric, then this will be the same as the mean. If not, your predictions will be biased. That's fine if you want it, but it should be kept in mind (Kolassa, 2020, IJF). The discussion at What are the shortcomings of the Mean Absolute Percentage Error (MAPE)? is related.