It sounds like there is a lot of tension and bitterness between you and your husband's ex. You may dislike her but this tension will eat away at your mental health, happiness of your husband and kids. No good will come of it.
The more you retaliate and get bitter, the more she will, and she'll try to cause problems for you.
You are blowing her request for crb check out of proportion. Its over-the-top, but she prob does not know you well enough to feel secure about leaving her child with you for weeks at a time. Yeah, she's prob requesting it and suggesting an accident happened to be mean. So, stop this cycle and don't rise to it.
Its stupid to make the comment that she could potentially ruin your career. If her accusations are unfounded, when investigated (unlikely she would go that far anyway), who's going to listen to a jealous bitter ex-partner?
Remain calm, just keep stating nonchalantly '"That's not what happened", as if you are totally unphased.
Your husband needs to deal with the crb issue. He's their Dad. He needs to wait for her to calm down and tell her calmly that she is worrying for nothing and its not going to happen.
Everytime you show her she's getting to you or behave in ways that show her you resent her existence, she thinks she is winning and so the poisonous cycle continues.