I'm fine either way, but I see no reason to overturn the way set-asides have been played since blue packs.
My only reason is that the set aside card doesn't SAY to send it with the character. And since the REG and rulebook don't either, there's really no reason to do it.
Basically we just have to decide whether we want set-aside enhs to work like:
1 - placed enhs and WC enhs that follow characters
2 - regular enhs that are discarded at the end of the phase they are played
I guess I'd rather avoid having a 3rd exception to normal enhs, so I'd lean toward #2.