Yes, 'cubic' is based on the Catmull-Rom cubic kernel. I agree the documentation for imwarp should be more explicit about this.
As for the "bias" in your code, I assume you mean an unexpected spatial bias? If so, I'd recommend working with floating point data and using 'linear' and go with a very simple geometric transformation like a half-pixel translation to narrow down the source of the issue. This should reveal where the problem is. You could go even simpler and work with a 1-D test "image" to really simplify things. I don't see how working with any of the cubic kernels makes a debugging issue related to interpolation easier since there are more neighbors to account for, but maybe I'm missing something about your particular problem.
Custom interpolants were a very rarely used feature of imtransform, so we did not include them in imwarp, but we could consider this moving forward. We could also consider other explicit cubic interpolation algorithms like B-spline moving fowrard.