The name change would have to be shown to be in the best interest of the child. I could see were an argument could be made that based upon the facts you have cited a name change would be in order here.
Daniel Bachert, Esq.
The Bachert Law Firm, P.A.
330 Clematis Street, Suite 222
West Palm Beach, Florida 33401
Please be aware and advised that this public forum is designed to provide only general information, to give you a basis of legal knowledge. This public forum does not give you attorney-client privilege. You and I have not entered into an attorney-client relationship. I am not responsible for your legal rights and this answer is based solely on the information you have provided in your question and as always, I would advise that you arrange for an in person consultation with an attorney from my firm or another Family Law attorney familiar with Florida Family Law who can analyze the specific facts and circumstances of your case more closely to better advise you.