Avoid involvement in banking and so not get on the wrong side of powerful monarchs?
1) Well, "banking" (in a primitive way) is directly due to their sucess. Having large lands, being trusted of goods, lending money to individuals, all comes first from gifts and such. You can't really prevent it, without damaging Templars' sucess first (but I don't give much about their lasting presence in Europe)
2) Templars Order dissolution is less due to them being huge lords and moneylanders. It was being a state within the state (critically with a king looking down on powerful feudal lords) at the point their possessions were technically sovereign alleus (more or less, totally on Templar's possession without suzerain)
For surviving Templars.
1) Latin States survives up to 1700. It's really unlikely. Even if you butterfly Hattin (by making Raimond of Triopli having the upper hand on how the campaign against Saladin would have to be led) there's too much menaces against them : Saladin, Mongols, Mameluks, Turks,...
There's plenty of Latin State survivales threads there, so you could take a look.
In this situation, Templars could remain the elite force of Jerusalem for longer (particularly if they don't know some decimation, Hattin-like)
2)No Trial of the Templars.
First you"ll need Philippe le Bel having a reason to keep them a thing. And seeing how he was the most powerful european ruler of its time (having puppetised the Papacy, set up roots of modern bureaucracy, etc.) he's not likely to do compromises.
So, Templars would have to submit and relinquish many things : treating king of France as their suzerain, loose total control on their property, being submitted to royal interests, loose their religious privileges (nobody was able to excommunicate them) and making Philippe (as he probably searched to) being Great Master of the order.
Basically, make them an Order of Santiago but even less autonomous.
You see the issue : Templars leaders (usually being great lords) aren't likely to submit. They didn't OTL and I don't see how they could have made it.
Even if it's reached, regarding their growing unpopularity (based on the fact their wealth didn't have a legitimacy, seeing how they couldn't defend Holy Land), their power...They're going to be absorbed and dissolved much sooner than XVIII century.
The best you could reach is a survivance of Templars' Order in a more or less anecdotal way (by example, as a formal order of chivalry) but they wouldn't be near of the "strong" part you asked.