None of the iberian military orders played any remarkable role in the exploration and colonization of the iberian powers. At the 16th century being a member of the Kinghts of Christ (rebranded actual templars under portuguese protection) Kinghts of Santiago, Knights of Alcantara or Knights of Calatrava (I think I'm forgetting some order) was pretty much a badge of honor and probably a way to have access to useful political and economical connections, something you added to your coat of arms and to your clothes and not much more. The hospitaliers continued having an actual function as military order due to their position in a Mediterranean contested between the ottomans and the catholic powers. My bet is that, with the right circumstances, a surviving Templar Order could play a role in the mediterranean dispute, but not in the New World, whose resources the Crown, involved in a proccess of centralization and legitimation of its power in the borning modern states of Europe, took care to monopolyze, even treating those who conquered new territories and became a problem, as traitors if necessary. Futhermore, the Inter Caetera was pretty clear. Of course butterflies could blow away the Inter Caetera, but then the history of exploration and first colonial experiences would be different to OTL.
All in all, I think that around the late 17th century this Templar order wouldn't be that different from the Hospitaliers and others, politically and military irrelevant (those functions being unfitting with Modern States that can cover that by themselves, unlike in the Middle Ages) and pretty much as said, a membership understood as a badge of honor, familliar tradition, whatever where mostly rich people play to remember old glories and make connections.Maybe they would keep strong links with the financial world, as suggested by others.