You can’t just do whatever you want with CVX; you have to follow its rules.
As for your 2nd paragraph, see 19. in p. 148 of “Lectures on Modern Convex Optimizatio”, Aharon Ben-tal and Arkadi Nemirovski https://www2.isye.gatech.edu/~nemirovs/Lect_ModConvOpt.pdf .
Perhaps you’ve read Computing or bounding a modified Ky Fan norm and Represent Schatten-p norm in CVX and https://github.com/nathanieljohnston/QETLAB/blob/master/kpNorm.m ?