If that's not enough to come back from an outage, you need to put in load shedding and/or back pressure. There's no sense accepting all the requests and then not servicing any in time.

You want to be able to accept and do work on requests that are likely to succeed within reasonable latency bounds, and drop the rest --- but being careful that an instant error may feed back into retry storms, sometimes it's better if such errors come after a delay, so that the client is stuck waiting (back pressure)

"Losing sleep" implies an actual problem, which in turn implies that the mentioned mitigations and similar ones have not been applied (at least not properly) for dire reasons that are likely to be a more important problem than bad QoS.

"Infrastructure" implies an expectation that you deploy something external to the troubled application: there is a defective, presumably simplistic application architecture, and fixing it is not an option. This puts you in an awkward position: someone else is incompetent or unreasonable, but the responsibility for keeping their dumpster fire running falls on you.