No you don't need to open up the engine all the way. It's just a push rod system, if the clutch cam isn't engaging with the push rod then it might be because the push rod has fallen beneath the cam arm so you're not able to position the cam behind the push rod.
I've never had this happen to me personally, I don't recall this being an issue, but the design is very simple so there is very little to go wrong.
What commonly happens is that the ball bearing inside the engine can get misplaced, which presents itself in a similar manner as you're describing. But this is not possible without dismantling the engine.
In essence what you have can be pictured as below:
|_._
The | is the cam that you slot from the top of the engine.
When the cable is pulled, it twists the cam which acts on the first pushrod denoted here by _. This is space separated with a ball bearing, denoted by . which pushes a further push rod denoted by _.
What might be possible is that you've ended up in situation where you've pulled the cam out and now the setup is like:
|
_._
The cam is now resting on top of the push rod and is no longer engaged with it. So when you pull the lever, nothing happens.
I can't remember whether this is even possible, but if it is then you'll notice the cam is sitting a lot higher outside the engine and would be obvious. I am not sure if the pushrods are under load, but if they're not, then leaning the bike over towards the clutch may cause the rods/ball to move allowing the cam to be lowered appropriately.
If that doesn't work, then no it's not a full dismantlement. It's just a case of taking the clutch cover off, and taking the clutch pressure plate off to expose the push rods. Extracting them (by leaning the bike over careful not to misplace the ball bearing) and then inserting the cam in first.
BUT I REALLY don't think that's what is happening here. It's probably more likely that the clutch just needs adjusting. I'd recommend referring to the Haynes manual for this.
