Hi,
So the field is of the table "user" or "address" I guess.
In that case, you want to edit the file "registration" (or "registration_bootstrap" if you have turned on the "use bootstrap option of the configuration) via the menu Display>Views and change the line:
echo $this->cartClass->displayButton($registerButtonName,'register',$this->params,hikashop_currentUrl(),' hikashopSubmitForm(\''.$this->form_name.'\'); return false;','id="hikashop_register_form_button"');
You'll want to put the call to hikashopSubmitForm(\''.$this->form_name.'\'); in a if statement with your ajax check so that the form is sent only if your check is valid.