Correct if I am wrong but Jews were huge supporters of the OE correct? In fact I believe Jews were the primary representatives of Ottoman Authority in the Levant. So what is the Zionist threat exactly? As long as the OE (which seems to be secular according to the OP) be threatened by Zionists? They allow a religious freedom and Islam has traditionally been friendlier to Jews than to Christians be concerned at all about Zionists?
Also one outside factor to consider, If WW1 goes differently and Germany comes out with a Status Quo Antebellum peace who is to say a WW2 even happens? Without WW2 its conceivable that there will be less agitation for a Jewish state.
The Jews of the Ottoman Empire were Asiatic and refugees from Spain for the most part, and were distinctly non-Zionist. They were highly favored and protected by the Ottomans, but I wouldn't call them the primary representatives of Ottoman authority, as they were very few in number. The influx of Ashkenazim from Europe were a different and more complicated matter.
Zionism as a movement was well underway by the early 20th c., and the Ottomans were ambivalent. It was mostly a cultural movement at first, so it was viewed as a welcome counterbalance to European imperial influence, but on the other hand, the territorial ambitions in Palestine of many Zionists were troubling, especially after Zionism in general had focused on the Holy Land around 1909.
In addition, remittances from European Jews to Palestine were good for the balance of accounts, and Palestine was no so overcrowded that additional taxpayers were unwelcome.
On the negative side, most insisted on retaining their Western citizenships, and too rapid an influx was destabilizing.
The former problem would have disappeared with WWI, as the Capitulations were abolished and most Ottomans lost their ability to retain foreign citizenship. The latter was a very difficult problem. Jews were welcome in any number into the Empire, but a limited number were allowed to settle in Palestine, although there was no limit on retirees. But once in the empire, you could make your way there, so the Jewish population was rising, albeit not to unmanageable levels.
So if Zionists made a concerted effort to colonize Palestine, it would lead to problems.
On the other hand, the Jewish millet had voted to drop all their various languages and adopt Turkish, so that would be pretty welcome if implemented seriously and an important bolstering of Ottomanism. But, the Jewish millet was dominated by non-Ashkenazi & non-Zionist Jews, so newcomers might be less enthusiastic about that.
Anyway, Zionism would certainly present challenges, but by itself, probably not a mortal danger to the empire.
Sorry, that was long.