Does Lucy feel trapped by her social class?

Yes, in Chapters 14-16, the exact nature of Lucy's social status is explained. Brontë spells it out ("her degree was mine," in Chapter XVI). Though Mrs. Bretton dotes on and patronizes Lucy, their "degree" is equal. That is, the social and economic status of Mrs. Bretton (who was formerly quite rich and the owner of a large mansion and enough property for a very comfortable income) is the same as that which Lucy was born into. Lucy's family (that great, barely-hinted-at, unknowable family of Lucy's past) must have been descended from some sort of gentry, or at least they were country landowners at some point. Lucy is not from any kind of working class; she was a full-fledged member of the educated, leisured, somewhat cultured upper-middle-class, English country gentry of the nineteenth century. For Brontë's contemporaries this state of affairs would make Lucy's fall into the status of a working teacher to be all that much more pathetic and even tragic. While the peril Lucy began her adult life in, both physical and economic, is certainly exciting and engaging to today's readers, the fall from one class to another has not the same sting it had in Brontë's age. The transition from one class to another–especially when downward, for a woman dependent on others’ income–was a subject not only of economic tragedy but of social and possibly moral degradation for those who suffered the descent. That Lucy, raised to be a lady at least of Mrs. Bretton's level, is now an English teacher in a Labassecourian girls' school is supposed to make her more sympathetic and tragic to the reader. The fact that modern ideas of social class mobility have changed would not necessarily influence the reader in the way a twenty-first century novel with a similar story would.