When combined to make a Large Chest, neither of the chest's names (when renamed in the Anvil) show in the GUI, it only shows "Large Chest".
Singular chests work fine, the name of the chest shows, but once combined, the name doesn't show. Both chests' names are still there if you break them and pick them up again.

I tried the following combinations (in every direction as well, in case it was something like "north-facing chests don't work". this is not the case, it is all directions.)

Unnamed chest, then named chest.
Named chest, then unnamed chest.
Named chest, then another of the same name chest.
Named chest, then a DIFFERENTLY named chest.

None of which had any name aside from "Large Chest" when combined. Upon removing one of the chests (making it a single-sized chest) the chest's name displays in the GUI.

I also tried placing the unnamed chest, then the named chest on the right side, as well placing an unnamed chest, then the named chest on the left. And vice-versa with named first, then unnamed on either side.

Pictures included are the names of the chests in the hotbar, the result of combining 2 chests into a large chest, and the aftermath of my testing various positions.