The fact that the answer isn't "move to RDS where Amazon solves the problem for us, which isn't our core business as a relationships app" seems to me to be a massive failing of the RDS offering and cloud services in general.
The RDS update process is a single button and you have no way of knowing how long it will take. There are some tricks like turning off Multi AZ and taking a snapshot manually before starting the process but still - for large instances you could be waiting anywhere from 30 minutes to 3 hours for RDS to finish. With large instance types I have seen RDS take a full hour just to provision an instance, in the meantime you'll sit there hitting F5 not knowing if it will ever finish.
Apparently Aurora behaves differently, but I wasn't aware of that when we specced out the project.