A rough idle could be the cam but it could also be something else such as a vacuum leak, wrong timing, bad plugs, bad plug wires, incorrect carb adjustment, etc.
Do you have a wideband on the car? What does the vacuum gauge say? What is the timing?
I'd be surprised if a cheater cam ran good in a street car since that isn't what they are designed to do, but I also wouldn't be surprised to find a bunch of other tuning problems in a new build. Every home built car that I've ever worked on has had a bunch of issues. That is just the nature of the beast.
Here's the problem- I bought the car last September and I have put 9 miles on it. I think no matter what it needs a tune. I'm definitely thinking that the cam needs to go. Man I love the sound of it, but it's just too rough- shakes the whole damn car at idle. If I was building a drag racing car, I'd keep it. But my goal for this is weekend cruiser.
The entire build has ~2k miles on it from what I was told by the previous owner, original block, heads.
So what would be a good cam for this car? I wouldn't mind a little lope/thump, but the current is excessive, especially being flat tappet. Aggressive ramps and flat tappet scares me.
Well I still don't think that 224-232-114 is a "rough" cam - the roughness could well be one of the many other issues folks have raised here - even a carb crudded up with ethanol/water sediment, etc. (Then again we aren't really SURE what cam you have in there, it could be quite a bit bigger than that, I am just ASSUMING those Lunati web specs)
BUT, if you really want a different cam, here are 3 that the factory (and I think most customers) thought were streetable. The durations are at .050 lobe:
206-220-116 hyd stock 428cj, also in 390gt, 428pi, and later in 351cj
A good clone for this cam is the common "RV" cam at 204-214-112
Another good clone is the Crower 270hdp at 210-220-112 (Werby's Wife's favorite cam)
220-230-116 hyd stock 429cj and 429pi
I ran this cam in my high school daily driver, and my girlfriend never complained. The cam didn't interfere with any of our planned activities.
228-228-114 solid stock 352/360hp, 390/375hp, 390/401hp, 406/385hp, 406/405hp, 427/410hp, 428/360pi, 289/271hp, 302boss, 429scj
This is the "AA" cam JDC mentioned above. A solid cam will idle a lot smoother than a hydraulic - it will act about 10 degrees smaller.
A good clone for this cam is the Comp 270s. I put this in my 88-year-old dad's build and he did almost 90 car shows with it, a real cruiser cam.
Any of these cams are GUARANTEED streetable cuz the factory said so & used em in many different engines, and they were all used in PI (cop) engines so they had to be pretty streetable.