From my limited non-professional experience, it sounds like a registration issue of the modems on TWs systems.
If up time is the highest priority here, the most direct route seems to be proving or disproving TW's "hardware issue" diagnosis. I'd pickup a TW rental modem and give it a try. If the internet still fails, then it will force TW to step up their efforts. If the modem provides steady internet then you have time to track down the error while maintaining service. The worst the client will face is a months rental charge.
My personal experience with owning cable modems is that Comcast offers terrible customer support for them, presumably TW is about the same. They will tell me my modem is properly registered on their system when it is not. The phone technical support say I must register the modem to the local store, where the rep will tell me it can only be dealt with over the phone. Documenting names dates and time of all conversations help in escalating the problem from a script reader to a properly trained tech.