Although I usually copy materials at Archives 1 and 2, I've done a few jobs at LC. There are some odd rules there, but I suspect that no one will be allowed to copy those documents - after all, if the docs are too fragile to be copied by their professional staff, it's unlikely they'd allow me to do the job. The finding aids you linked to had a place to contact their staff - ask if I (or anyone else) would be allowed to make copies and what equipment would be acceptable. If they say it's OK, contact me and we can work out the details.
Cheers,
Dana