Populism is defined as anti-intellectual philosophies that go against established norms and offer solutions to the problems of common people that go contrary tradition. There are only two characters in Pride and Prejudice who might come even close to this definition; they are Georgiana's former governess and Wickham. The former governess, Mrs. Younge, only fits the definition in that she engaged in behavior that went contrary to tradition by making arrangements for Georgiana to run away with Wickham. However, this behavior would more readily be called unscrupulous rather populist. Wickham only fits the definition in that he goes against the established norms by leaving debts all over every town he inhabits and by running off with young women. Again, this is more readily called unscrupulous and immoral rather than populist.
Elizabeth is a gentleman's daughter. In that era, this meant that she was the object of some privilege and social standing. Had she had an opportunity in Pride and Prejudice (if the storyline had been different) to do so, she could have gone to London during the "social season" (season of theatrical performances, parties, balls, concerts, etc.) and socialized with the best families, including nobility, which means that if she--and her small fortune--could attach the affections of a gentleman of London's high society, she would have been socially eligible to accept his proposal of marriage.
The above is said to explain that Elizabeth--though living in the country and having a small fortune to be given her upon her marriage--is a member of the power elite. As such, she is expected, and expects for herself, to associate with equal and higher members of the power elite. She also expects and is expected to marry within the power elite. Her education, speech, conversation, leisure, interests all mark her as a gentleman's daughter and therefore as part of the power holding classes. She would virtually never have an opportunity, and probably no inclination, to associate with or identify with the populist individuals or groups that have anti-intellectual philosophies that contradict tradition and go against established norms.