Two reasons immediately come to mind as to why Charlotte would insist Lizzy play at Lucas Lodge (6)—to entertain the company and to exhibit so that everyone, even the eavesdropping Darcy, can admire her style and ability. Entertaining and exhibiting go hand in hand and I think was expected of accomplished ladies at social gatherings. Lizzy, I don’t think, really has an objection to playing and takes the opportunity to tease Darcy about his fastidiousness saying “I would really rather not sit down before those who must be in the habit of hearing the very best performers”. It occurs to me that like Sir William’s future gallant attempt to bring Lizzy & Darcy together by a dance perhaps Charlotte is doing the same by the way of music—I have heard music soothes the savage beast even the ones to whom dancing “is a compliment which I never pay to any place if I can avoid it”. (:D)

Charlotte seems to be a natural at manipulator. Prior to insisting Lizzy play, she had asked Charlotte “What does Mr. Darcy mean… by listening to my conversation with Colonel Forster?” and also said that “He has a very satirical eye” suggesting she has also observed him watching her. Perhaps Charlotte thinks Darcy might be interested in Lizzy and like him that his silent observation and listening is only a precursor to conversation thus she takes it upon herself to gently nudge them into the next level of socializing.

On his [Darcy] approaching them soon afterwards, though without seeming to have any intention of speaking, Miss Lucas defied her friend to mention such a subject to him; which immediately provoking Elizabeth to do it, she turned to him and said -- (Ch. 6)

When Darcy came near again (above) Charlotte dared Lizzy to confront him which provoked Lizzy to do it and then before they speak four lines to each other Charlotte addresses Darcy saying it “It will be her turn soon to be teased… I am going to open the instrument, Eliza, and you know what follows” - Charlotte is telling Darcy he will have his opportunity to tease or rather talk to Lizzy when she is finished playing. What Charlotte has done is this: She provokes Lizzy into talking to Darcy and then puts Lizzy’s natural talents on display for his admiration but reminded Darcy to talk to Lizzy afterwards. Perhaps Darcy cherished this expectation but Mary’s eager impatience for display thwarted his ambition and then when Mary continued after her own exhibition was over by purchasing “praise and gratitude by Scotch and Irish airs” thwarted him again and that is why:

Mr. Darcy stood near them in silent indignation at such a mode of passing the evening, to the exclusion of all conversation, and was too much engrossed by his own thoughts to perceive that Sir William Lucas was his neighbour, till Sir William thus began -- (6)

Remember that Charlotte suggested Jane “had better show more affection than she feels” (5) in order to secure Bingley and then when “she is secure of him, there will be leisure for falling in love as much as she chuses” (5). Considering Charlotte’s views on courtship and marriage Lizzy’s lack of cordial feelings for Darcy may not appear to her an incurable obstacle to a match so by showing off her lively friend to the gentleman, who she believes “has a right to be proud” (5) she might believe she is doing them both a good turn. Is Charlotte an unromantic matchmaker and Lizzy & Darcy her victims? (:D)

Unless otherwise noted all quotes are from Ch. 6. (:D) Excerpt of Lyrics - from Fiddler on the Roof: