Are there any early industrial age methods capable of determining diamond or graphite's crystalline structure?
Would 1850's level spectroscopy or other technology of the time be able to demonstrate the crystal structures of carbon (specifically graphite and diamond) without a 1900's understanding of atomic structure?


Yes, using visible light and petrographic microscopes. The field of optical mineralogy was mature enough by the mid-1800's for one to determine the crystal structure of minerals. In particular, the work of Sir William Nicol is worth investigating (thin but useful Wikipedia article).

I will add that there's a pretty simple way to determine whether or not the crystal structure of a mineral is the same or different than that of another mineral - the so-called crystal habit.

Finally, diamond exhibits conchoidal fracture, and graphite does not. This might be the easiest way to tell that the two minerals have different crystal structures.