I think you can introduce a new variable z and add the constraint norm(x) <= z, then use @Michal_Adamaszek 's solution from How can I write this kind of constraint in cvx .
Putting it altogether, and in order to deconflict variable name, substituting t for x in my code for that solution in the last post of that thread, we get, if I haven’t made a mistake,
variables x(2) y z s t
norm(x) <= z
{z,s,y} == rotated_lorentz(1)
{s,t,z} == rotated_lorentz(1)
Then use t in place of norm(x)^3/y^2