You answered why in your post:
Bree had no intention of letting all the crowd know that he could talk.
A horse breaking away from the crowd and running right up to a person in the procession would no doubt be suspicious. And also, people would be able to see him looking like he was talking and the Narnians talking back even if they couldn't hear what was being said.
Also, what good would that have done? The Narnians couldn't have taken Bree with them without people seeing the "great war horse". At this point, the Narnians were not yet expecting to sneak away. No doubt they would have been sent off with fanfare and trumpets and a procession down to the docks when they set off. Someone would be bound to recognize that Bree was a war horse and there might even be someone who would recognize whose horses it was.
On top of that, it is not said whether or not the Narnians brought horses with them but I'm pretty sure they didn't. They would have anything they needed provided by the Calormenes. And a horse is not exactly something you can sneak on board a ship easily.
Say Bree did run up to the Narnians and the Narnians decided to try to buy him back. How would they accomplish that? If I were a Calormene, I would be thinking
what is so special about this horse? Why do they want it so badly? There must be something that makes them want this horse so badly and would start probing to try to find the answer. If the Narnians said he was a talking Narnian horse, you're back to square one and the Calormenes wouldn't sell him because they'd be able to make tons of money off of him. I personally would not buy a story of someone just "taking a fancy" to that particular horse.
Even if the Calormene's sold him without knowing he talked, how would they get him back to Narnia? They can't take him on the royal ship. Someone would have to ride him across the desert and back to Narnia. Seems like a lot of pains to take for a horse, even a good horse. I would be suspicious and wondering why they're taking such pains for a horse when they probably have dozens in Narnia and can buy a bunch at once from somewhere and have them shipped on a livestock boat whenever.