The correct carburetor for that engine is a Keihin BB62. I can see from your picture it is a Keihin BB, so I doubt it is the wrong one. The BB does have its idle channel, and its adjustments, on the opposite side compared with the BE used on the GXV (pushrod) engines. See illustration:
[Linked Image]

When you cleaned the carburetor, did you clean the idle jet? It is located below the vertical screw-in plug just forward from the idle mixture screw (which is locked to prevent adjustment)? It is not removable, so it is a bit tricky to clean. Even more importantly, did you remove the emulsifier, which is above the main jet? Both of those parts are very vulnerable to fouling on that carburetor, and moderately difficult to clean. The symptoms seem quite a good fit for a gummy emulsifier, and it cannot be cleaned in place, it has to be removed from the carburetor.

The compression test you have described is only relevant to side valve Briggs engines, and is used because the Briggs EasySpin feature makes a conventional compression test meaningless. You should do a compression test on that Honda with a gauge: specified pressure is 71 psi.