In my opinion, Lyddie feels obligated to support her family. I do not believe that Lyddie ever feels love or joy at the fact that she is helping to support her family and pull them out of debt.
Lyddie feels slightly betrayed at the beginning of the book when her mother forces her to work at Cutler's Tavern to help pay off the family debt. A big motivation for Lyddie to leave that particular place of employment was to earn more money, and there is always a sense of urgency that she has to work harder, longer, and faster in order to earn more money. She isn't saving much of it for herself. She simply wants the debt paid off as quickly as possible.
When her sister is dropped off in the city, Lyddie isn't exactly overjoyed at having her sister there with her. Lyddie feels it is an inconvenience to have to go to work, find some employment for her sister, and figure out how her sister can stay with her. There is a general sense that Lyddie feels burdened by her sister. Lyddie also knows that it is her responsibility. That is why I generally feel that Lyddie's main attitude about supporting her family is an attitude of obligation rather than heart-felt love.