Metric to punish positive errors
How about using the difference of an exponential and a linear or a polynomial functions?
For example, if $x = actual - predicted$, then
$$L = w_1 (e^x-1) - w_2 x$$
The cost of error increases faster for positive values than for negative ones, and is a differentiable function of $x$. You can tweak $w_1$ and $w_2$ depending on your typical errors and your learning algorithm. This is an example, for $w_1 = w_2 = 20$: