That is totally possible. I think the way they did it with a Vue app in an i-frame loading sketchfab in another i-frame and communicating via Sketchfab viewer API is a bit cumbersome. I think you would have a better solution directly with babylonjs. What they are doing to change the logo is a set texture call - you can have that positioned with UV mapping.
Yes, totally, checkout this 3D Material configurator with editable Texture transforms (rotate/offset/scale/flip/etc.). Upload any Texture file on the fly, or even combine different textures (the site currently is under development, so you may get errors while playing with the material editor):
Custom Logo is just a matter of uploading transparent logo as Texture file and not repeating the texture. Or use Babylon Decal.
However, if you try to mix GLTF files from right handed system with Babylon built-in meshes using left handed system, there are issues currently.