As long as you are traveling w/ a child, you are allowed to check a stroller at no additional fee. This is usually done at the gate. Due to the size of the Chariot, it might be a better idea to check it at check in.
Are you bringing it w/ the jogging or stroller attachment? Maybe if you are using the jogging, it might not be a bad idea to pack the wheel in your suit case so it doesn't get lost.