In which layer does the 'becomes the color of your choice' part of the Wild Mongrel effect apply?

The 'becomes the color of your choice' effect of Wild Mongrel applies in Layer 7c, which is the layer where color-changing effects occur. In Magic: The Gathering's rules, effects that change a permanent's color are specifically handled in Layer 7 because this layer is solely dedicated to changes in color.

When Wild Mongrel’s ability is activated, it modifies the color identity of the creature. This happens after any previously defined colors have been established (which occurs in earlier layers) and before any other effects that depend on the creature's color are applied.

Layer 5 addresses changes in power and toughness, Layer 6 deals with effects that modify the base values of creatures, and Layer 1 is for the creation of permanents, none of which specifically pertain to color changes as described in this specific ability. Therefore, Layer 7c is the appropriate layer for resolving the Wild Mongrel's color-changing effect.