Would Silver Flame Templars have standing to take action in Breland or Aundair? By take action I mean particularly "martial" actions, such as capturing a person (as recognized by the treaty of Thronehold) and taking him for an exocism, or killing a demon disguised as a citizen of Aundair. Would these types of actions cause a diplomatic incident?
Well, not officially. Some SF zealots don't care about diplomacy and will go to any extreme justified by their beliefs.
In 832 YK, there was an incident called the Silver Crusade, which was a lycanthrope purge carried on by the Church of Silver Flame. As Aundair was a location very plagued by werewolves, this made the Silver Flame quite popular in Aundair, specially around the Reaches, but no one forgets easily the ties of the Church with the Thranish government.
So, the SF priests have no legal standing in Aundairian or Brelish territories, but some slight might be forgiven if there were proof that it was a demon purge, or something like that. Parties that were deeply involved in the Last War might be influenced by their involvement with Thrane in the war. In example, an Aundairian expatriate from Thaliost would not look kindly upon SF acts.
In the novel The Queen of Stone: Thorn of Breland, by Keith Baker (Eberron's author) there was an interesting discussion over the Silver Crusade and its consequences. It is also possible to see that Silver Flame Templars do see themselves as "saviors" and they tend to take action if they see demon/lycanthrope/undead activity.