You mentioned that charis does not mean "undeserved kindness", I agrre with you, however if we look at the phrase in question there is a word that indicate a free gift. That word is:
G1432
δωρεάν
dōrean
do-reh-an'
Accusative case of G1431 as adverb;
gratuitously (literally or figuratively): - without a {cause} {freely} for {naught} in vain.
So I would have to disagree with you on the part where you said there is not free gift in the passage. Now then we have to look at ourselves, are we worthy of such a gift? If not, then the gift would be undeserved would it not?
G5485
χάρις
charis
khar'-ece
From G5463 ; graciousness (as {gratifying }) of manner or act (abstract or concrete; {literal} figurative or spiritual; especially the divine influence upon the {heart} and its reflection in the life; including gratitude ): - {acceptable} {benefit} {favour} {gift} grace ({-ious}) joy {liberality} {pleasure} thank ({-s} -worthy).
The word gratuity from wikipedia :"gratuity", dates back either to the 1520s, from "graciousness", from the French gratuité (14th century) or directly from Medieval Latin gratuitas, "free gift", probably from earlier Latin gratuitus, "free, freely given"