"Batman shouldn't be forced to kill the Joker at the detriment of his morals and mental health" AND "Bruce himself set up Jason's expectation that parents will take bloody vengeance for the sake of their children's lives" are statements that tell you Bruce is in a hell of his own making.
Jason does in fact have the right to ask. Bruce gave it to him implicitely. Because Bruce a) positioned himself as Jason's father, b) blamed Jason for Garzonas' father trying to get revenge on Batman and Robin and dying for it, c) argued that it was the natural actions of a father to try and avenge his kid.
Either Bruce IS Jason's father, and he failed to live up to the standard he set, or he isn't, and he sold a lie to an orphan to put him in the line of fire to deal with the loneliness of his eldest leaving the nest.
Jason, when coming back to life, seeing the Joker alive, must have felt incredibly betrayed and foolish. "I thought I'd be the last person you'd ever let him hurt."
Jason's ultimatum in Under the Red Hood is asking Batman to put his money where his mouth is.
Was it the natural thing for Garzonas senior to set-up an ambush to gun down Batman and Robin because fathers should avenge their sons?
Was it really Jason's fault back then that three men died in a failed scheme? When Batman toppled the pile of cars on top of Garzonas senior, when he tricked the subordinates into shooting each other, can we still blame Robin for not seeing this coming when he 'maybe' pushed Garzonas off the balcony?
If it is so inevitable that more deaths would follow a son's murder, why is the Joker still alive?
And if it wasn't, if fathers don't normally take bloody vengeance, then perhaps the three men that died should be blamed on the man that 'pushed' them?
Maybe Batman already had blood on his hands.
Maybe the Red Hood has a right to ask why he's so comfortable with that.
