I'm Running Out of Things to Call These

Firstly, Caroline attempts to accuse Elizabeth of being a "pick me" girl

“Eliza Bennet,” said Miss Bingley, when the door was closed on her, “is one of those young ladies who seek to recommend themselves to the other sex by undervaluing their own; and with many men, I daresay, it succeeds; but, in my opinion, it is a paltry device, a very mean art.” “Undoubtedly,” replied Darcy, to whom this remark was chiefly addressed, “there is meanness in all the arts which ladies sometimes condescend to employ for captivation. Whatever bears affinity to cunning is despicable.”

But back to Caroline. I don't think Caroline fits the modern definition because she does not set herself apart by claiming to be different from other women, her argument seems to be more that she's An Ideal Woman. Her criticism of Elizabeth's muddy walk is, "I, a proper woman, would never do that (neither would your well-mannered sister)." She tries to demonstrate that she's more elegant than Elizabeth by walking around the room together. We know that she has a fancy education and many accomplishments. We never see her play herself off as a tomboy or engage in male activities.

Mary Bennet is probably the strongest candidate for a "pick me" girl:

To this, Mary very gravely replied, “Far be it from me, my dear sister, to depreciate such pleasures. They would doubtless be congenial with the generality of female minds. But I confess they would have no charms for me. I should infinitely prefer a book.” (emphasis author's)

but she also doesn't have that aspect that engages in male pursuits.

Some people accuse Elizabeth Bennet of being a "pick me" girl because of this quote:

“My beauty you had early withstood, and as for my manners—my behaviour to you was at least always bordering on the uncivil, and I never spoke to you without rather wishing to give you pain than not. Now, be sincere; did you admire me for my impertinence?” “For the liveliness of your mind I did.” “You may as well call it impertinence at once. It was very little less. The fact is, that you were sick of civility, of deference, of officious attention. You were disgusted with the women who were always speaking, and looking, and thinking for your approbation alone. I roused and interested you, because I was so unlike them. Had you not been really amiable you would have hated me for it: but in spite of the pains you took to disguise yourself, your feelings were always noble and just; and in your heart you thoroughly despised the persons who so assiduously courted you.

However, Elizabeth was not trying to attract Darcy at all, she was borderline rude and acted unlike other girls because she genuinely disliked him. This was not a tactic.