Mixed Integer SDPs which are not Mixed Integer SOCPs

I do not think I can be quite categorical about this. CVX will indeed convert all 2x2 LMIs into SOC equivalents (both real and complex). And CVX does not use 3x3-or-larger LMIs unless it has to. But if you force the issue by specifying, say, an arrow-shaped 4x4 LMI, CVX will not detect that.