Different Bane actually. It's not even his real name.That was a lot of putting stuff together, but there are some major flaws.
First, nothing in the Doomspace write-up says "Dawn War." There could have been 10 million wars between different gods and across the cosmos.
Second, Asmodeus, Bahamut, Tiamat, Lolth, Gruumsh, Moradin, Sehanine and Corellon are not Forgotten Realms gods. They are gods that are multi-setting. It isn't FR that has absorbed them. It's they that have absorbed FR and other settings. The only FR god on your list is Bane.
Third, even if it was the Dawn War, the 5e gods involved do not have to be the same ones as 4e. Each edition makes changes to lore.
In short, if you WANT it to be the Dawn War and you WANT to involve FR gods, you can make it that way, but the default as of right now isn't that way. The default is mysterious war involving mysterious gods and mysterious primordials.