The hunt for gollum is a hard quest. Bilbo isnt the best hero, but he has his uses, and with the up comming sets, it looks like hobbits will be getting some nice boosts (they already have access to fast hitch and dwarven armor which are both great cards. Fast Hitch + Dwarven armor + bilbo + burning brand is a great combo).
As for the quest, yes, you can beat it with that card pool. I beat it using just the core, THfG and CoTC cards. KD has some great cards which should mean you can definately do it.
The reason the quest is so hard (without spoiling) is because a few special abilities are key in order to really have any chance of beating it. If you don't have decks that have access to these abilities (which you will have access to them from your current card pool, it just depends on whether you build the right deck or not) then you can get slaughtered pretty easily.
That being said, it can be a fun quest, its just a really specific quest that requires a specific build… which means that you unfortunately have to really focus on building around it.
If you want more detail just let me know and I can go into more specifics. Have fun.