LD1:
Your rifle seems like a fine piece. And it may, indeed, have no number. However, I'm willing to bet that basic disassembly will reveal a four-digit number somewhere, and on a number of parts. The reason for such a number was to keep the gun's bits organised and together during build up -- as well as to record the piece in company archives. Otherwise, your original guess of 1883 or so seems not out of reason.

Best, Kensal