You mean $\int K(x,y)^qd(x,y)<\infty$? For which $q$ and how is that dependent from $\Phi_k$ and $u_k$? Just to make clear what I mean: In case $\Phi_k(u)=|u|^{p_k}$ a simple sufficient criterion would be the finiteness of the mixed norm $\int\left(\int K(x,y)^{p_2'}dy\right)^{p_1}dx$ which is already rather different than the first criterion.

Edited the last condition so that the result becomes more general, and the growth condition is of Krasnosel'skij type.

Could you please define what you mean by "Kantorovich conditions"? There are numerous sufficient boundedness conditions for positive integral operators between weighted Orlicz spaces in literature (but no necessary and sufficient ones). Also the term "locally inferable" does not seem to be so standard to me.

As already observed in a comment here, this argument shows only that the definition of "more" (in the sense of existence of an injection and lack of a bijection) does not make much sense without choice. Something which is not really surprising given that many sets have no cardinality (in the sense of cardinal numbers) without choice. If looked at it this way, the argument only boils down to the fact that $\mathbb R$ is such a set if all sets are Lebesgue measurable.