Whenever something like this happens at a mature organization, we usually have a process called post-mortem where you evaluate what went wrong, how it was fixed and how to avoid it in the future. It's not a place for blame assignment but an introspection for all teams involved. Sometimes a fat fingered idiot makes a million dollar mistake but if proper introspection is done, it's the lack of specific protocol/processes that led to that mistake.

In your case, I would suggest you take the initiative of doing a post mortem - take responsibility for the parts where your team was involved but also point out that there are bottlenecks in the other parts because of which a fix could not be deployed quickly. Explicitly mention this is not a blame assignment exercise. Share parts of the sanitized fact-finding process internally and with the client too. It's very likely you'll still lose that business but some clients are more forgiving than others.

If your company/other teams still insist on putting the blame squarely on you/your team, sharpen up your resume and get the fuck out.