One possible solution might be having the client introduce an artificial delay of 10 seconds or some other time when it encounters an expired cert, or adds an additional second of delay for every day it is expired. This degrades the connection but does not immediately break anything.
Plus you'd need to be way in the guts of the TLS implementation to achieve this; if you're already there, start generating noise a week ahead of the expiration instead.
Or better, none of the above and automate.
We wanted a database server to fail hard. Running slowly just caused cascading failures.
Of course, in this case you're effectively talking about the entire cluster crashing hard, but that's still easier to cope with than every system responding at a snail's pace.
The goal of a business is not to have perfect engineering practices. It is to fulfill customer requests. When there is an outage in the middle of the night, I'd argue that a degraded system buys time to address the issue.
Regardless of the mechanism, having a sudden, complete breakage is not ideal for a business.
If you plan to implement something like this, then do it right and have the service catch the exception and notify an administrator.