If the fact that Doge has no real development and is based on a dog meme didn't kill it, I don't think anything will.
It has a real world use... primarily moving funds between exchanges and I suppose gambling sites. And as others noted, it has a ton of liquidity. It's a joke coin of course, but at this point I expect it's a joke that will last as long as crypto does.
BAT looks good too and I like the real world usage aspect. However, KYC for the brave browser probably will limit adoption a great deal. I don't see BAT and Doge as direct competitors.
That is the major use of doge to me as well, for transfer purposes.
I have never used BAT til now for that purpose.
And as you said, because they require KYC for the brave browser, people will be limited on this network.
They have evolved into different applications in the crypto market.
So in conclusion, BAT will never kill the almight doge.