Will discharging work if the grounding chassis isn't connected to the mains or otherwise grounded?
Yes, this is the only(!) correct way of doing that. The CRT itself acts like a plate capacitor (and a very good one at that). The charge is stored between the inner parts of the CRT and its outer conductive coating - not between the CRT and earth. If you short the inside of the CRT (via the anode cap) against its outside (via the various grounding strips connecting to the outside of the CRT), it'll get discharged just fine.
If you instead shorted the CRT's anode against earth, but not against the metal chassis of the monitor, it would not get discharged. The monitor's chassis instead would be at a very large negative voltage, which might potentially be quite dangerous.