I just thought of another fringe possibility for the Templars to survive as a holy order. And that would be to sell out to the muslims. The problem is that there would be very few muslim states at the time that would either see a use for having a Christian Holy order under their command, and even fewer that would be acceptable for the Templars to join.
The only valid candidate would be the Ottomans once they capture or are about to capture Jerusalem, but that'd be a long period of time the Templars would need to manage and not be destroyed in.
If they did manage to survive up untill then, maybe, just maybe they'll start working with the relatively Pro-Christian Ottomans in exchange for getting to manage Christian Holy sites in the Levant. This would however mean that the Ottomans can't give these sites to the Orthodox Christians, which would give the Turks a less strong relation with the Church as in OTL, and all they'd get in return would be a semi-loyal Christian Holy Order, that may go back to their original business of fightiing against Islam at any moment.
So if the Turks were to ally the Templars (an allready unlikely possibility) they would certainly do away with them as soon as the threat of the Mamluks would be gone, and the Ottomans would have no use for a group of infidel zealots that help defend the Holy Land, as they would be a bigger risk than they're worth.
So basicly there would be no way for the Templars to continue their existance as a Holy order up to the reformation, no matter what they'll try.
If they stick around in the Christian world, they will be wiped out by any Pope or King that has the power and reason to do so,
and if they switch allegiance, they'll just end up betrayed once they become a liability.
A holy order is allways a liability, in the Christian world too, that's why they were eventually destroyed in our timeline, and it's what will happen in any other timeline as well