I was orginally just looking for a way to play a campaign as my limited understanding of Descent 2E was that it is was still single scenario based. I wasn't looking to adapt RtL specifically.
Descent 2E was actually designed with campaign play in mind. It's not the same as in RtL (it's a bit more streamlined with some more freedom of progress), but the base game actually assumes a campaign progression across a number of quests, and has a small section of rules for one-off play. (This is the opposite of the base D1e game.)
The Shadow Rune campaign contains 20 different quests, most with 2 encounters, and for a given play-through of a campaign, you'll play 9 of those quests (the first is always the same, 3 of 5 Act I, 1 of 2 Interludes, 3 of 10 Act II, and 1 of 2 Finales). To play through all of the quests in campaign form, you have to run the campaign at least 4 times.
If you're on a budget and don't want to pick up the full game, since you already have the rule and quest books, conversion kit, and dice, you'll at the minimum need to pick up the small cards (classes, items, and conditions), and the big cards (travel, monster, and overlord).
What this doesn't include:
D2 Heroes (sheets and figures): You've got Conversion Kit heroes from 1e, so you could play with just those.
D2 Monster Figures: You'll need to use proxies for these.
Map Tiles: You could theoretically just print your own or draw the maps on a standard gridded playmat since you have the quest guide.
Tokens: You can easily print or use proxies (coins?) for these as well.
That said, if you're not on a tight budget, I would just go ahead and pick up a full copy of the base game. Of all the things that it's good to have spares of, dice are always a good idea, and rule / quest books are also pretty handy so that you can have one for the hero players and one for the overlord player. (FYI: All quests are considered public knowledge in D2, with only a few details being listed as exceptions.)
Cheers and best of luck!