I'm not 100% sure if this is the answer I'm looking for. If someone has a comment to confirm, or a better answer, please feel free. I'm not really sure what he is saying; it sounds more like the place fixes the character traits than the Leviim who are there, but one could be a byproduct of the other.
I found the Shem MiShmuel (the son of the Avnei Nezer), says something like this based on an idea from the Kotzker Rebbe (the Avnei Nezer's father-in-law).
הנה ידוע שמקומו של אדם למטה הוא שורשו גם למעלה, וכבר הגדנו זה בשם כ"ק זקיני האדומו"ר הגדול זצללה"ה מקאצק, וע"כ ערי הלוים שורשם נמי כמדת הלוים, וע"כ חבורת לוים שהיו דרים במקום אחר זולת עריהם, אינם קולטים מפני **שהמקום גורם...**
Behold it is known that the place of a person below is also his root above, and we were already told this in the name of my grandfather the Kotzker Rebbe, therefore the cities of the Leviim also had their roots like the characteristics of the Leviim. Therefore, a group of Leviim who would live elsewhere, outside their city, wouldn't protect [the accidental killer] because the place itself causes [the protection]...
The end of this paragraph seems to negate the idea that it's the Leviim who change the person, rather it's the place. But he's saying something about the "root" coming from the Leviim.
וע"כ ערי מקלט שהופרשו מהן יש בהן נמי רק חיות ממעלת המדות ולא החיות ממעלת השכל, וע"כ אינן נותנין חיים רק למי שפגם במעלת הגוף והמדות דהיינו שחטא בשוגג שלא נפגם השכל אצלו אלא החטא והפגם הוא במה שלא הי' מצד עצמו נמשך להיפוך משפיכת דמים, כי השלם בגופו ובמדותיו, אין האברים שלו נמשכים מעצמם אף בלי הוראת השכל לדבר שאינו ראוי...
Therefore, the cities of refuge which were separated, they had a lifesource from the virtue of character traits, and not a lifesource from the virtue of the intellect. Therefore, they only gave life to someone who blemished the virtue of the body and character traits, that is that they sinned accidentally and not that they blemished their intellect. The sin and the blemish was that they didn't pull themselves away from bloodshed, since when someone who is perfect in their body and character traits, their limbs aren't propelled on their own to something inappropriate, without the command of the intellect...
This paragraph sounds like part of the idea, that a person couldn't have sinned this way if they had perfected their character traits.
וכן נמי במי ששפך דמים בשוגג הוא פגם בגופו ובמדותיו ולא בשכלו, ע"כ לאיש כזה הוא תיקון ערי המקלט ששורשן בחיות הגוף והמדות
Similarly, someone who spills blood by accident has blemished his body and character traits and not his intellect, therefore a person like this will be fixed in the cities of refuge, whose root is the lifesource of the body and the character traits.
Again this sounds like the place and not the people.