How to set zero of bicolor ColorData?

You could make your own bi-linear mapping:

myColorFunc = (ColorData["GreenPinkTones"][
               Piecewise[{
                          {Rescale[#, {-30, 0 }, {0, 1/2}], # < 0},
                          {Rescale[#, {0  , 10}, {1/2, 1}], # >= 0}}]]) &;

BarLegend[{myColorFunc, {-30, 10}},LegendLayout -> "Row"]

Another way to shift the position of a middle color of a gradient uses the "bias" function of Schlick (with suitable rescaling) along with a color gradient. Here is a small demo:

bias[h_, x_] := bias[h, x, {0, 1}];
bias[h_, x_, {a_, b_}] := a + (x - a)/(1 + (1/h - 2) (1 - (x - a)/(b - a)))

Manipulate[DensityPlot[x, {x, -30, 10}, {y, 0, 5}, AspectRatio -> Automatic, 
                       ColorFunction -> Function[x, ColorData[{gradient, {-30, 10}},
                                                              bias[h, x, {-30, 10}]]], 
                       ColorFunctionScaling -> False, FrameTicks -> {Automatic, None}, 
                       PlotLabel -> Row[{"h=", Round[h, 0.001]}]], 
           Row[{Control[{{gradient, "GreenPinkTones"}, ColorData["Gradients"],
                         ControlType -> PopupMenu}],
                Control[{{h, 0.5}, 0, 1}]}, Spacer[20]]]

where we see through experimentation that setting h = 0.25 shifts the white color to 0. To confirm:

Solve[bias[h, 0, {-30, 10}] == bias[1/2, Mean[{-30, 10}], {-30, 10}], h]
   {{h -> 1/4}}

Thus, here is the final color function you need:

mycf = ColorData[{"GreenPinkTones", {-30, 10}}, bias[1/4, #, {-30, 10}]] &;
BarLegend[{mycf, {-30, 10}}, LegendLayout -> "Row"]