[KY] Custody between Kentucky and Florida

My son is 7 year old, autism (level 3), ADHD and type 1 diabetic. My son’s father is not on his birth certificate so I know I would need to establish paternity. I plan on asking for child support because I just can’t do it any longer. My rent has increased from $1,100 to $1,475 in a matter of 5 years and I’m sure it’ll increase this October when it’s up for renewal. I’m exhausted because I am the only one who takes care of my son and have to take him to various therapies and ensure his blood sugar doesn’t go high, fall and I’m up constantly at night to ensure that as well. It’s hell on earth.

I would like to ask for child support but I’m scared he will try to ask for child custody when he has no idea all the care that my child entails. He’s an eloper, has hypoglycemia unawareness and is nonverbal. His father is currently married to the woman he cheated on me with (she has no idea my son even exists). If my child didn’t have special needs I wouldn’t care to ask for child support but my son has so many needs and I think it’s only fair that he pays something. I don’t want him to have custody but would he be granted it being that he’s never been around and my son has two major disabilities? Just curious and thank you in advance.