Per 901.15 a custodial arrest can be made for a felony or misdemeanor committed in the officer's presence, and for any felony for which there is probable cause to support the arrest.
But, I don't know why the focus is on a minor child? Aggravated Stalking per 748.048(3) indicates the following:
Any person who willfully, maliciously, and repeatedly follows, harasses, or cyberstalks another person, and makes a credible threat with the intent to place that person in reasonable fear of death or bodily injury of the person, or the person’s child, sibling, spouse, parent, or dependent, commits the offense of aggravated stalking, a felony of the third degree, punishable as provided in s. 775.082, s. 775.083, or s. 775.084.
No requirement for a minor exists.
So, I suspect the officer charged and arrested for a felony based on the information they believed existed at the time. A court reviewed the information and found that there was insufficient probable cause to support the charges ... at that time. Understand that the charges can still be filed if there is enough evidence to support them. And, apparently, they will still have to face the charges for a first degree misdemeanor per 784.048(2):
(2)Any person who willfully, maliciously, and repeatedly follows, harasses, or cyberstalks another person commits the offense of stalking, a misdemeanor of the first degree, punishable as provided in s. 775.082 or s. 775.083.