A more formal word for the verb "bagging"
To avoid sounding too informal, the OP could use "bagged" in a passive construction e.g.
Cash and cheques are bagged and sent off via…"
If the perfectly acceptable verb bag still bothers the OP, possible alternatives could be "packed", "packaged", or "closed/sealed in bags"
Cash, cheques and other valuables are packed and sent off via…