Would I be wrong to call in to my nursing job?

I'm an RN at a very large hospital. We have been very understaffed lately. I try to be as helpful as I can and even have stayed later to help. I NEVER call in. However my son has an evaluation on the 5th for an autism evaluations. He starts school on the 26th and I'm trying to get this evaluation done before the school year starts. The place that does them has an extremely long wait list. And by long I mean I made this appointment in November and August 5th was the soonest they could get us in. I requested off in December and it got approved last month but Tuesday my day off request was changed from approved to denied. When I asked why I was told that because we have had so many people quit she just doesn't have the staff to let me off. I told her I need this evaluation so we can get a diagnosis and hopefully get my son an IEP. He's in middle school and this would really help make his school year easier. She said there's nothing she can do. I called asking about the possibility of rescheduling and they said they wouldn't be able to get us in until June but I could be on a wait-list for cancelations. So I would be called if there is a cancellation but over 40 people are on that wait-list and I would be at the bottom. I'm thinking about just calling in. She doesn't have the staff to fire me and even if she did I would have another nursing job like that. I know we are supposed to be there for our patients and there's stuff with ratio and all that but I am a mom first. And in reality.... I'm entitled to my PTO. Them not having coverage isn't technically MY problem. Normally I would never even consider this ... But I have been waiting MONTHS for this appointment and she denied it so last minute.