If you attempt to pickpocket the advisor and fail, he will attack you.
Isran asks the Dragonborn to find and destroy the vampire masquerading as a Visiting Advisor to the Jarl of a radiant city; however, to avoid slandering the Dawnguard's name, permission must be gained from the Jarl to kill the creature or the guards will see it as murder. You can … A Jarls Justice.
If this occurs in front of other NPCs, they will then attack the advisor.
You’re tasked with locating a Vampire pretending to be a Visiting Advisor to the Jarl of a randomly picked city.
I have tried everything to pickpocket him successfully (light clothes, potion of invisibility, muffle enchant on boots, ring of sneaking +30% and potion of pickpocketing) but I cannot pickpocket him successfully.
If the jarl has given you permission to kill the Advisor, your bounty will not increase, but you will still be pursued for any bounty you already have. I am trying to complete the quest A Jarl's Justice where you have to pickpocket the Visiting Advisor to get the orders to show to the Jarl (The quest sent me to Dawnstar).
It seems that your next quest would have been "A Jarl's Justice", and that this quest completes automaticly after a change of rulership. If you do kill the vampire without permission, you will complete the quest, but you will have to face the consequences if you are caught doing so.
Isran will give you this Vampire-killing order (Gunmar must have been too busy for this one). The devs just seem to have forgotten to check if it is the current quest.. ;-) This also should work (but requires testing) with Season Unending.