There is a "reload textures" button down at the bottom right. I think you have missed that, because that should definitely reload an updated skin, along with any other updated textures.
I know it's not ideal. My current plan is to add a viewer mode in the LFS garage which would be a lot like the CMX Viewer and you could look around any mod you have installed in there.
It's really important to do as soon as possible but we have had to go for this release without further delay.