Unfortunately, these types of poison kill the mice/rats by interfering with the clotting cascade...meaning that they bleed to death because they're unable to form clots. The same thing happens when your dog ingests the poison. I noticed that in your question, you stated that the vet said an anticoagulant would counteract the rodenticide, and I'm hoping that they didn't give her an anti coagulant, but rather a coagulant...if they gave an anti-coagulant, it likely made the problem worse accidentally.
The coughing can indicate that she is getting worse, so the best thing to do is to at least give your regular vet a call and see if they think she needs to be seen. If they do, then they'll get her started on a regimen of IV Vitamin K, rather than the oral, which binds with the rodenticide and helps with the clotting factor. They'll also run some blood work to make sure that her kidneys and liver are not being affected and check her clotting time to judge how much damage has been done to her clotting cascade. They may want to start her on an IV to flush her system a bit.
With early intervention, these dogs do well...so I'll keep my fingers crossed for the positive outcome!