zecher to the machtzis ha'shekel -- 3 coins or 1 -- Rama vs GR"A

The Rama (O.C. 694) writes that the minhag is to give three machtzis ha'shekels (bad dikduk, I know) on Ta'anis Esther because the word terumah appears 3x in parshas shekalim.  However, GR"A in Ma'aseh Rav (282) writes that the minhag is to give only one machtzis ha'shekel.

Aside from trying to understand the nekudas ha'machlokes between them, it's very hard to understand the Rama.  The reason we read parshas shekalim is to remember the annual donation of machtzis ha'shekel given to the mikdash to pay for korbanos tzibur expenses.  We once a year do something to remember this once a year donation. The additional shelakim alluded to in parshas shekalim were one-in-history donations used in constructing the mishkan.  Why do we need an annual remembrance for that?

My son pointed out that the GR"A in S"A on this Rama points you to Tos in Megillah (21) which says that on ta'aneisim we give tzedaka at mincha time.  My son suggested that the GR"A is perhaps not just explaining to us when the machtzis ha'shekel is given, but is defining for us the geder ha'din, what machtzis ha'shekel is -- it's not a zecher, but a din in hilchos tzedaka, just like tzedaka given on any ta'anis.  Therefore, the shiur is machtzis ha'shekel, not 3x a machtzis shekel.  The Rama understood that it is a zecher.

I'm not yet convinced that the GR"A there is doing anything more than explaining why the machtzis ha'shekel is given on ta'anis esther afternoon -- why not give it sometime close to when we read the parsha?   I would formulate the issue a little differently.  The GR"A sees our machtzis ha'shekel as a zecher to the machtzis ha'shekel of the mikdash.  The Rama perhaps sees giving machtzis ha'shekel as part and parcel of our kiyum of parshas shekalim.  By way of analogy, RYBS and others understood that our eating matzah on pesach night is not just a kiyum in achilas matzah, but is a kiyum in sipur as well -- it's part of how we tell the story.  So too here, perhaps Rama means that reading parshas shekalim also entails donating 3 machtzis shekalim --- davka 3, because 3 are alluded to in the parsha, and our giving is in some way an extension of the mitzvah of reading the parsha.  (I don't know if my explanation is any less dachuk, but I have no other ideas yet.) I found that R' Yisachar Shlomo Teichtel hy"d, who these days is better known as the author of Eim haBanim Smeicha, discusses this topic in his shu"t, Mishne Sachir.  (O.C. 34).  He quotes what appears to be two contradictory gemaras: On the one hand, the gemara (Meg 13) quotes Reish Lakish as teaching that the shekalim that Klal Yisrael gave in the midbar served to nullify the money Haman paid to bribe Achashveirosh.  We proved with our pocketbooks that we are dedicated to good before he even had the idea of showing his financial dedication to evil.   On the other hand, Chazal tell us (Meg 16) that Haman chanced upon Mordechai teaching hilchos kemitza of the korban mincha and remarked that the small offering of kemitza outweighed all the thousands he had poured into doing evil (Meg 16).  So which was it -- was it the shekalim donated to the Mishkan, or was it the actual offerings of the Mikdash, e.g. kemitza -- which foiled Haman's plans?

In a simplified nutshell, these two views represent the roots of the GR"A vs Rama.  According to Reish Lakish's view, the "kesef ha'kipurim" donated to the mishkan served as a kaparah not just for that generation, but for future generations as well -- for Mordechai and Esther's time, for our time, for all time.  Therefore, according to Rama we make a zecher even for the machtzis ha'shekel donated for the building of the mishkan.  However, if Haman's downfall was brought about by hilchos kemitza, it is the korbanos themselves, purchased with the single machtzis ha'shekel given annually, which is what saved us, and therefore, as GR"A writes, it is this single machtzis ha'shekel that we make a remembrance for.