Purpose specific tokens make sense when the scope is "limited enough". The public does actually use laundromat tokens, food tickets at fairs, arcade tokens and similar things. They wouldn't take them seriously outside of the appropriate context but they definitely take them seriously within them.

Sure, but the world isn't a Chuck-E-Cheese's restaurant. The real world deals in cash, not tokens to be exchanged for trinkets. It stretches the imagination to believe that website purveyors are going to want to deal in Brave Bucks, or whatever they want to call them. And if they can be exchanged for arbitrary goods and services, as opposed to a tightly constrained set of options, it's going to be deemed a currency.

What if the token were only redeemable on-network for advertisements and reading, and otherwise the only thing to do with it would be sell it on an exchange for currency?

We're going in circles here.

Here's the thing: the law looks to intent. If the sole purpose of issuing these things is to evade regulations by coming up with something that's not designated a currency but otherwise operates like one, no amount of jumping up and down and screaming "it's not a currency! don't call it a currency!" isn't going to make it something other than a currency. Courts aren't dumb and they don't look kindly to parties who try to game the system.

In your hypothetical, the existence of an option of selling this thing on an exchange for other currencies makes it very similar to any other kind of currency. I think you'd have to eliminate that option to avoid getting too close to the line. But if you don't have that option, I just don't see how it'll have any significant value to the recipient.

Most non-currency things can be sold for currency. What makes BAT more similar to a currency than those things?

I'm totally speculating here, since I'm not intimately familiar with the pertinent regulations, but I think it's not going to be about the token's inherent properties as much as the mechanisms surrounding it.

With a token such as a laundry/car-wash token, an amusement ticket, or a Disney Dollar, you can't get cash back from the issuer. Once you buy them, they're yours forever, unless you can find a third party to give you money for them. The fact that you can (try to) sell them on eBay to a willing recipient doesn't make them currencies.

So I think it comes down to who controls the exchange. If it's a third party with no connections whatsoever to the issuer, then it's unlikely to be considered a currency. But if the issuer is also operating the exchange, or has a connection to the operator, then I think it's going to look a lot more suspect in the eyes of the law.

Again, this isn't legal advice - consult a licensed attorney in your jurisdiction.