I figured out another way to pick colors from a bitmap when using the Mesh tool. You can't directly use the eyedropper and paint bucket to apply colors because the mesh becomes deselected when you use those tools.
Also, the mesh becomes opaque and you can't see your bitmap underneath once you make a mesh.
Here's my workaround:
1. Open the Color docker.
2. Make a copy of your bitmap and move it over to the side. Then set up a second view window that's focused on the duplicate bitmap.
3. Use the eyedropper to pick a color from the duplicate bitmap. This will be selected in the color docker.
4. Switch to your original window and select the mesh with the mesh tool.
5. Select your node or patch on the mesh, and click the "Fill" button on the color docker (which has the color you selected from the bitmap still selected).
I should make a feature request about this. It ought to be possible to use the eyedropper and paint bucket to select colors while leaving the mesh active. But for now, this is the only way I've been able to figure out. If there is an easier way, I'd be happy to hear about it.
Hope that helps!