Maybe. I assume the two of you were never married, so you'd be filing for paternity to invoke your father's rights and establish timeshare.

There are many ways to establish temporary attorney's fees or a "loser pays" rule in Florida. The most common way is to file your case and show to the judge that she makes a lot more money than you. Another way is to show that she is being a jerk in the litigation--such as refusing to file proper paperwork or cooperate in scheduling. Another way is to offer a certain settlement using a certain legal form and she refuses but you win the case. You should discuss these options with a lawyer.