The main argument of the Shebbatai's disciples was that his apostasy, his conversion to islam, as well his death was supposed to happen.
Sabbatai‘s followers were instructed to reject the halakha and used mystical reasons to justify their position by explaining that the rejection of the mitzvot was a key step in messianic redemption, as Gershom Scholem says:
The fundamental conception of the Zohar, the Bible of the Kabbalists,
is that in the time of grace, in the world of order (Olam ha-Tikkun),
the laws of Judaism, the regulations concerning lawful and forbidden
things, would completely lose their significance. Now this time, the
Sabbatians thought, had already begun; consequently, the minute
ritualistic code of Shulchan Aruch ought no longer to be held binding
(Scholem, Sabbatai Sevi: The Mystical Messiah, p. 142. Princeton University Press)
According to this (expounded by one of the most zealots sabbatians), Samuel Primo (c. 1635 - 1708):
"...they had to adopt a radically different attitude toward the values
that had been dominant until then, namely the the Law of Moses and the
halakhic tradition of rabbinic Judaism". (Scholem, ibid pp. 797-798. ).
"... disobedience to the Jewish religious law became acceptable and
even encouraged in the sect while purporting to have a desire for
holiness, and a belief in God and the Messiah. Abandoning tradition
seemed to legitimize and even encorage assimilatory thinking as a
religious tenet... (Avrum Ehrlich, Encyclopedia of the Jewish
Diaspora, p. 779-780. ABC-CLIO Publisher).
His Conversion to Islam
Many of his followers thought that his new (muslim) name was a sign that he had not really converted. One group took from the Lurianic Kabbalah the notion of Shevirat ha-Kelim (or "breaking of the vessels") claimed that his conversion was not a betrayal of his faith but "a decisive and unique mission" that would allow him "to enter the realm of the kelipah in order to destroy it". Prof. Dr. H. Graetz in History of the Jews, Vol. V, 1975, p. 158. Cosimo Classics) by mentioning the belief of his followers (under the guidance of Nathan of Gaza, 1643–1680) says:
"It was a Kabbalistic mystery which some writings had announced
beforehand. As Moses was obliged to reside for some time at Pharaoh‘s
court, not as an Israelite, but to all appearances as an Egyptian,
even so must the last redeemer live some time at a heathen court,
apparently as a heathen, ― outwardly sinful, but inwardly pure".
The frenzy of his disciples did not ended with the death of Shabbatai. Many of his adherents looked upon his death as a mere "retirement" or "concealment". Some of his followers expected him to return as messiah later.
The sabbatians too believed that the redeemer's absence (a moral
absence after his apostasy, a physical absence after his death) was
temporary only and that he would return before long to achieve his
messianic mission (Scholem, ibid p. 796).
In a letter written and disseminated by Nathan of Gaza (see above) his return was explicitly foretold:
A year and a few months from today, he (Sabbathai) will take the
dominion of the Turkish king without war, for by the power of the
hymns and praises which he shall utter, all nations shall submit to
his rule. He will take the Turkish king alone to the countries which
he will conquer, and all the kings shall be tributary unto him, but
only the Turkish king will be his servant. (...) "In the seventh year
the son of David will come" (San. 97a). The seven years that is the Sabbath
signifying king Shabetai; At that time the aforementioned rabbi will
return from the river Sambation, together with his predestined mate,
the daughter of Moses...mounted on a celestial lion. (Scholem, ibid pp. 273-274)
Hope that helps.