Normally it's pretty obvious how to switch between approaches and skills, but this one has me stumped. The cliffhanger is so intimately tied to the number and nature of approaches that I'm unable to hack it into a useable and coherent form with skills.
Well, my first suggestion is to instead of using easy way, less easy, tricky, hard, ect, is to steal from 4th DnD's skill challenge options as they are very similar. Require X successes, but Y of them need to be this skill and you can only use Z skill once. Everything else can set up aspects but not count as success.
There is also the Rodrigo method, from Critical Hits. His homebrew is that you cannot roll the skill you rolled last turn and you cannot use the one the player before you just used. I've used the successfully in multiple systems, including Dresden. Here's a more detailed explanation and an example from the podcast starts around 24:30 and goes till 39:00 but the wrap up is a little longer. Some background in case you skip right there, it's a D&D adventure, party almost wiped last session in a tower fighting skeletons. The wizard alone survived by retreating and has come back in to try and rescue his friends.
To give a personal example, I ran a Dresden game where the players needed to get from New York to...I believe it was a white council meeting? Not sure. Anyway they had to do it fast and since they had a wizard they thought planes would be a bad idea. They went through the never never and ended up in a Fae mapping of the Labyrinth, from Greek legend. Instead of drawing a maze, I ran it as a skills challenge. They needed to get, I think it was 5 successes, before 3 failures. I narrated the area of the labyrinth they were in, a hedge maze, looked familiar, to the point they were sure they were there before. They were stuck in an endless loop. Wizard rolled evocation to burn through the hedges and they exited to a new area, where there was a giant venus fly trap. Werewolf distracted it with agility, nipping at the plant, while everyone got by. They then came across a set of statues, and the champion of Odin tried to figure out what they were using lore. He failed and they grabbed him. I forget the next two failures but they ended up getting stuck and the winter knight appeared with the Minotaur to slaughter them. In order to survive they jumped out of the never never without looking, ending up in the Hudson bay I think.
Using this method it kept people engaged and the requirement to switch skills demands that people don't always default to their best, keeping things varied and interesting. As you can see, the basic mechanic, roll X success before Y failures is the same. The idea of approaching things the wrong way still applies. Trying to use fire on the hedges was easy, however trying to know about a random set of statues in an old and mythical place was very hard. The labrynthin is meant to be mysterious and unknowable, so lore was always going to be difficult, similar to the hard vs easy approaches. It's quick and simple because you don't need to map anything out. Just set the outline of the challenge and react to the players ideas. The only place it becomes more complex is that skills are more specific than approaches and as such isn't quite as elegant though slightly more elegant than the base Skill Challenge from 4th. Though I suggest reading up on advice for those anyway. You may find that specificity of some examples actually maps better for certain scenes than the vaguery of a cliffhanger.
To maintain that elegance both example loss quickly, you should use just use Modes as your categories. Modes perform a very similar task to approaches, in grouping skill actions into a larger more inclusive category. However this still doesn't seem great as some Modes don't map well to the "how" style of approaches. So instead of modifying Cliffhangers, why not modify your character sheets to include approaches per the conversion guide?
Cut the problem down to (about) six allowed skills, and go from there.
Approaches are much wider in scope than skills are. In a broad scenario like a cliffhanger, it's not usually possible to totally rule out approaches. As ineffective as Forcefully breaking down the foot-thick iron doors sealing you in the death trap might be, at least it's conceivable. But how would you have any impact on that scenario with, say, Resources? Empathy? Vehicles, unless you brought one?
Well, you wouldn't. And you don't have to. Any more than the GM is obliged to set you a difficulty to purchase the sun, jump to the moon, or glare at gravity until it lets go of you. It doesn't waste your turn to "try" something patently impossible. There is no try; it just doesn't happen in the first place.
So, if you want to run a cliffhanger in a skill-having system, here's how you do:
1. Decide allowed skills.
What could possibly work? If you get up to five or six that's probably good; people are going to have about half the skills in the game on their pyramid, if you're in Core or something on a similar power level like Robo. The odds that none of them show up for somebody are quite small, especially if you think about approaches when you're going down the skill list so you don't just cluster all the Banter skills into the allowable bits.
2. Spread difficulties.
Since you're not going off exactly six skills there isn't the neat 1-2-2-1 diamond, necessarily, but do try to have at least one skill at each tier.
3. Run it!
Trying to use a skill with no listed difficulty doesn't make you waste a turn or roll at +9 or something. It's just, that doesn't work, what else you got? Just approach it with the same open-mindedness as the original cliffhanger - in an example of play, someone decides to be Flashy in the deathtrap and play dead, because Dark Stobolous is totally watching and will deactivate it so he can come down and gloat in person, which is something the GM hadn't thought of and shifts the Flashy difficulty down from +7 to +3. If somebody tries a skill you didn't think was possible, ask them how they think it's possible, and if it makes sense, judge difficulty accordingly and let them at it.