The 93cM does add to the hypothesis of Thomas being James’s Father.
If HD is willing to give you Management rights to their DNA and if they are also willing for their DNA to be uploaded to other sites then you could do it.
I talked my Cousin through assigning me the Management of his DNA and I now have his Raw DNA data on my PC which I am going to give to him when I send the family tree to him on a memory stick. Hence I could upload it to other sites for him, but as yet we have no need as his DNA unlocked some mysteries on his other line.
Do remember that at 2xGGP or more likely 3x and beyond that it is possible that no DNA is actually inherited by persons currently testing or recently tested.