The chelbe'na symbolizes "Poshei Yisrael," and RChbBiz in Krisus 6 says that a community fast that does not include sinners is not a fast at all.
Krisus 6b
Menachos 27a by the four minim on Sukkos
Sanhedrin 37a by Yitzchak's bracha to Yaakov where it says וירח את ריח בגדיו.
There are four approaches to that statement.
1. They are sinners, but they still join us in the fast, and we should not reject them. (Rashi here)
2. They are sinners, but they still join us in the fast, and they play an important part in our community so we should welcome them. (Drashos HaRan Drush 1.)
3. This are only welcome when they are sincere about their tefillos, and want to do teshuva, but they are not strong enough to really do teshuva, so our prayers will join with theirs and enhance their Teshuva. (Maharal, his neighbor the Kli Yakar, and their contemporary the Mabit, and the Chida.)
4. This are only welcome if they already did teshuva. If they didn't, we don't want them. (Rabbeinu Gershom in Krisus, the Maharsha in Krisus.)
Another interesting fact about the Chelbenah for Purim:
We all know that Mordechai is alluded to in the fragrances that comprised the shemen hamishcha in Mor Deror, translated as "Mira Dachya." This Mor was also used in the Ketores - the Mor that we mention in Pittum HaKetores, which is among the other ingredients that are not explicit in the Torah. (The Ramban/Malbim in 30:34 say they are hinted at in the word "Samim" that begins the Ketores, which means "take the samim mentioned above that were used in the Shemen HaMishcha, and also....." However, the Avnei Nezer holds that the Mor (cholem) in the Shemen HaMishcha was musk from the musk ox and the Mor (kametz) in the Ketores was a plant derivative.) Not everyone knows that the Chelbe'nah, in the Ketores, is the same gematria as Haman. Off the cuff, this might be reflected in the fact that Haman's grandchildren were talmidei chachamim. Alternatively, it might be because it was Haman's threat of extermination that led to Kimu v'Kiblu. And, of course, many baalei machshava have said that the former is related to the latter. (מאדים is the same gematria.)
Krisus 6b
א"ר חנא בר בזנא א"ר שמעון חסידא כל תענית שאין בה מפושעי ישראל אינה תענית שהרי חלבנה ריחה רע ומנאה הכתוב עם סממני קטרת אביי אמר מהכא (עמוס ט, ו) ואגודתו על ארץ יסדה
A similar thought is expressed in two other places.Menachos 27a by the four minim on Sukkos
ד' מינין שבלולב ב' מהן עושין פירות וב' מהם אין עושין פירות העושין פירות יהיו זקוקין לשאין עושין ושאין עושין פירות יהיו זקוקין לעושין פירות ואין אדם יוצא ידי חובתו בהן עד שיהו כולן באגודה אחת
Maharsha there
ד' מינין שבלולב ב' מהן עושין פירות כו' וב' מהם אין עושין כו'. כל דברי המאמר מפורש בפיוט יום ראשון דסוכות וכמו בעץ הדר ריח וטעם כן כו' וכמו הם אגודים כו' ולכפר אלה על אלה כו' ע"ש באורך דהיינו ב' כתות מהן שעושין פירות דהיינו מעשים טובים נדמו לאתרוג ולולב שעושים פירות ושנים מכן שאין עושין פירות ומעשים טובים נדמו להדס ולערבה דאין עושין פירות וצריכין להיות אגודה א' שיגינו צדיקים על רשעים וק"ל:
Sanhedrin 37a by Yitzchak's bracha to Yaakov where it says וירח את ריח בגדיו.
סוגה בשושנים שאפילו כסוגה של שושנים לא יפרצו בהן פרצות והיינו דאמר ליה ההוא מינא לרב כהנא אמריתו נדה שרי לייחודי בהדי גברא אפשר אש בנעורת ואינה מהבהבת אמר ליה התורה העידה עלינו סוגה בשושנים שאפילו כסוגה בשושנים לא יפרצו בהן פרצות ריש לקיש אמר מהכא (שיר השירים ו, ז) כפלח הרמון רקתך אפילו ריקנין שבך מלאין מצות כרמון ר' זירא אמר מהכא (בראשית כז, כז) וירח את ריח בגדיו אל תיקרי בגדיו אלא בוגדיו
There are four approaches to that statement.
1. They are sinners, but they still join us in the fast, and we should not reject them. (Rashi here)
2. They are sinners, but they still join us in the fast, and they play an important part in our community so we should welcome them. (Drashos HaRan Drush 1.)
3. This are only welcome when they are sincere about their tefillos, and want to do teshuva, but they are not strong enough to really do teshuva, so our prayers will join with theirs and enhance their Teshuva. (Maharal, his neighbor the Kli Yakar, and their contemporary the Mabit, and the Chida.)
4. This are only welcome if they already did teshuva. If they didn't, we don't want them. (Rabbeinu Gershom in Krisus, the Maharsha in Krisus.)
Another interesting fact about the Chelbenah for Purim:
We all know that Mordechai is alluded to in the fragrances that comprised the shemen hamishcha in Mor Deror, translated as "Mira Dachya." This Mor was also used in the Ketores - the Mor that we mention in Pittum HaKetores, which is among the other ingredients that are not explicit in the Torah. (The Ramban/Malbim in 30:34 say they are hinted at in the word "Samim" that begins the Ketores, which means "take the samim mentioned above that were used in the Shemen HaMishcha, and also....." However, the Avnei Nezer holds that the Mor (cholem) in the Shemen HaMishcha was musk from the musk ox and the Mor (kametz) in the Ketores was a plant derivative.) Not everyone knows that the Chelbe'nah, in the Ketores, is the same gematria as Haman. Off the cuff, this might be reflected in the fact that Haman's grandchildren were talmidei chachamim. Alternatively, it might be because it was Haman's threat of extermination that led to Kimu v'Kiblu. And, of course, many baalei machshava have said that the former is related to the latter. (מאדים is the same gematria.)
וזהו הענין הנרמז בחלבנה הנכנסת בסמני הקטורת, כי עם היות ריח החלבנה מצד עצמה בלתי נאותה, אפשר שיהיה לה כח לעורר ולהוציא איכויות הסמים האחרים ולעורר בשמיותם.
It would be nice if we could find a connection between the Ketores and Esther, but all I can think of is the fact that she was perfumed for a year in several of the ingredients used in the ketores, a pretty poor connection - ששה חודשים בשמן המר וששה חדשים בבשמים. Or that she was called Hadassa, a fragrant plant, but it is not an ingredient of the Ketores.
Alternatively, we could say that Esther was not merumaz in the Ketores, but she is in the daled minim, in the Hadassim. True, the Haddas is a symbol of maasim tovim without peiros, but, unfortunately, that was Esther's fate as well, considering that her children were not part of Klal Yisrael. So both Mordechai/Mor Dror, and Esther/Haddas, are present in places where they are bound up with people without any zechusim of their own, the Chelbenah and the Aravos.