That whole interaction suggested she was being sarcastic.
*Dude pops up at her office unexpectedly with flowers*
Joan: I thought about it, and I'm going to send my son away so he won't be a nuisance for you to deal with
Dude:
I didn't ask you do that
Joan: yes you did
Dude: look, I'm sorry, I'm trying to be less rigid, so although this wasn't in my plans, because rigidity makes you old.
Joan: I also live with my mom's
Dude: idc, I want all of you in my life
She's clearly throwing his objections in his face. She lied about not having a son in LA because she was content on letting it be a west coast fling. He showed up in N.Y. on his own accord, to get surprise. While she was upset on the burden her son was having on her social life, and sending him away is certainly not beneath her, they never show her actually preparing or thinking about doing that. She didn't start looking at boarding school brochures or asking someone in the office about boarding school.
Also, her son is, what 4 years old? Not many, if any, boarding schools start that young. She said it so bluntly like that cuz any sane person would know it was too ridiculous to be serious.