Dear Christian,
Glad to hear the Great Eastern site is progressing, and sorry to hear about the dearth of authoritative documentation. I'm afraid I dont have any information regarding GE hull painting, but I have noticed in photos of large liners a diversity of hull paint prior to, and sometimes after launch, the grey repainted black by sea trials. The Mauretania was grey with white upperworks to trials; Queen Mary entirely grey at launch, while Queen Elizabeth lanched in black and white, and so on. I also note variances in the position of white superstructure - Lusitania had a white band to the bow, painted black by maiden voyage. All of this is to say that shipyards and shipping companies had variable ideas about protecting the hull during construction phase and later in "dress" for service.
The best advice I could offer is to reseach the types of marine paints of the Great Eastern's era, their composition, etc. Publications like Engineering and so on may be helpful here, as well as Scientific American or its British counterpart.
As if a 680 foot 1858 ship was not hard enough.
Good luck!
Jim