I agree with TomÃsimo; it's weird to use "despachar" or "facturar" in Mexico in this context. The first might be used if you were the employee who must send the luggage to the baggage porters and it would still feel strange for me. "Facturar" we only use it for tax paying.
Here, we normally say "documentar el equipaje / la(s) maleta(s)", but "registrar" is fine.