For slow responses, if you have to use something like Microsoft Teams, set your status to "busy". Corpos will see that and think, "Oh, right, they must be busy because the little red icon says so!"
These people are really by the book, so if the software tells them you're busy, you must actually be "busy".
As for missed deadlines, that one is difficult. I used to miss deadlines with stuff all the time because my previous job was really bureaucratic, so if something needed to be taken care of by "the tech department", nothing would actually get done and the deadline would get missed. You best bet is to just find a way to pin it on your manager, like by saying that the manager didn't "manage" you well you enough and that's why you missed it. This works unless your manager's manager (jeez...) is the one who hired them, in which case, they'll defend your manager and blame you anyway. You know, because if their hiring choice is bad, then it makes them look bad.
Ugh... those environments...