Short Answer: No, Dumbledore is not one of the 3 brothers.
Long (Interesting) Answer:
When we see the Tale of the 3 Brothers, what do we notice in each brother?
The Eldest one wanting power, thus asking for the Elder Wand.
The Middle one wanting to raise his dead love, thus the Resurrection Stone.
The Youngest one wanting peace and to not see death again for a while, this Death’s Cloak
Let’s see which people we know would fit into this….
Voldemort. Wanted power, sought the Elder Wand, thus leading him to be the Eldest brother
Snape. Wanted to honor his dead love and bring her back, thus being the Middle brother.
Harry. Wanted peace, thus being the youngest brother.
Now that we’ve identified the brothers…. Who’s Death?
Well, if we look deeper into the story. It says that the youngest brother greeted Death as an old friend.
So… after Harry was hit by Avada Kedavra (The Killing Curse), who did he meet at King’s Cross….
Who did Voldemort take the Elder Wand from? Dumbledore.
Who did Snape follow while trying to help his dead love? Dumbledore.
Who gave Harry his Invisibility Cloak? Dumbledore.
Now, it also says that Death was cunning, wasn’t this proved by all the plans Dumbledore formed leading to the end?
So, in conclusion, we can see Dumbledore not as a brother but as a guide, a guide akin to which death was to the brothers in the story. A guide to the brothers in reality.
Edit 1: Under concerns of Plagiarism, I’m stating that I did not make this theory up. I found it and believed it would be interesting to inform others about it. However, I have added a few points of my own. If you believe it is plagiarized please contact me.
Basically, though I didn’t originally think of the Dumbledore being Death theory, this is my answer. Furthermore, I have added quite a few points of my own as well.