I haven't gotten many Bernie supports just C + B with Byleth and C + B with Felix (currently playing Golden Deer). I'll be sad to find out that other characters make fun of her anxieties in supports if that's the case. Also haven't played Black Eagles so I will concede that it may feel more forced and disingenuous when she's there at every group chat and important plot moment but as someone who has been anxious for as long as I can remember, I never felt like she was funny or meant to be comedic. I could see how others might think the fact that it gets brought up or shown in nearly every interaction is a sign that it's being played for laughs but based on personal experience I can say that when anxiety is that severe it really does begin to envelope you and show up all the time.
Thankfully, I don't have any experience with the kind of abuse Bernadetta received and our anxiety stems from different places so I'll defer to those that do for any final judgment on the portrayal. I will say that from her background she seems like someone who would internalize more than externalize and we do get some indication that she does but due to the need for outwardly spoken dialogue and reactions I am not sure how to handle it better than they do.