I like the advice from CW and R2C. I would also suggest you do not overthink things. I think you had a pretty good handle on parenting before the D, right? So why change things drastically after D?
I also do not see a problem with you offering to get your daughter help. I probably would have worded it differently than you did. "Working on it" in her mind may have meant to her you thought you guys should go after breakfast and have her pet a dog. So be clear:
"I understand you feel embarrassed because of your fear of dogs. If you would like, at some point when you are ready, we can look into resources that might help you with that fear. We can look over options like counseling, etc, and if you are comfortable with any of the options we can look into it."
Let her be in charge of the if, when, what, where and how. That why you are supportive, but she gets to decide.
Scott, you are a good guy doing what you feel is best for your kids. I have documented my strained relationship with my daughter here before. Our relationship is really good now. I wasn't perfect, but I tried to always do right by my daughter, and now that she is in her late teens she is starting to see that. So do not judge your relationship with her in a specific point in time. Let her see you constantly keeping her best interest in mind, and eventually she will come around.
M(53), W(54),D(19) M-23, T-25 Bomb Drop - Dec.23, 2017 Ring and Piecing since March 2018