Well, they did in a sort of way : the suppression of the Ordrer didn't physically threatened Templars that weren't in the kingdom of France, and most of the fortune (even in France, most of Templars's holdings went to Hospitalers) and people there joined up with Hospitalers, or iberian orders.
Now about the survival of the Order, that's going to be more tricky. Templars were totally discredited popularly for being basically bankers and having gave up to the apparences the protection and reconquest of Holy Land, as well monacal life.
Having the pope not compying before Philippe le Bel is going to be hard as well, considering the success the french king already had on papacy.
It should be pointed, however, that the Order wasn't suppressed then its rulers arrested, but the reverse. Indeed, it seem that Clementius V tried to protect them and the order was
dissolved, rather than suppressed, afterwards.
The easy answer would be a delayed fall of Acre and remnant of Crusader States, but it would only delay the fall of Templars (and maybe before this TL Siege of Acre).
What could save Templars, would be both to still have holdings in Eastern Mediterranean (as Hospitalers), to keep far more religious discipline, and to be less successful as an order in western Europe.
Templars problem, again, was once Holy Land out of Christian control, they had no point anymore; while Teutonic or Hospitalers were present "on the front".
Having a less important unifying tendency in France at this point, or at the contrary that Papacy turning even more deeply under french influence would help a lot (as Templars under pontifical alliegence rather than royal would seem less of a threat).
Of course, having pope renouncing on Temple's suzerainty and giving it to the secular rulers would be a good thing for the OP, but I don't think it would be really doable.
Eventually though, Hospitalers and Templars are going to be quite redundant, and one is going to absorb the other. Giving the more maritime-minded nature of Hospitalers, and their easier adaptation to the loss of Palestine, I'll still bet on them.