In the meantime, I would edit online checkin to show the absolute latest checkin time possible (something like 5am the calendar day after the start date), and keep trying.
I have had luck NOT mentioning a reservation number during the phone process, then once you get someone on the line tell them what you need (to let them know you WILL be there but will be delayed, and to hold the room) and THEN mention the actual reservation. Something like that isn't a revenue-based thing and really shouldn't be under the control of the TA, so if you start off with a clean slate (no reservation for them to see) and with exactly what you need, you'll have a better chance at a meaningful conversation with the CM than if you start with reservation info.
And just keep in touch. They know that people are traveling. They know delays happen. Flight delays, train delays, car accidents, etc. Keeping in touch with them is KEY.