This is a more general version of my previous question (Yet another question on finite fields), which has yet to attract any comments (nevermind answers). It occurred to me that I might be getting bogged down in the specifics. So, here goes:
Suppose I have two distinct maximal proper subfields $K$ and $L$ of a finite field $F$.
Let $\alpha \in K^*$. When does an element of $\{\alpha N_{F/K}(x) : x \in L^*\}$ have order less than $\# K^*$?
If $K$ and $L$ are distinct maximal proper subfields of a finite field $F$, then there must be a prime $p$, two primes $q_1\ne q_2$, and a positive integer $r$ such that $\#F^* = p^{q_1q_2r}-1$, $\#K^* = p^{q_1r}-1$, and $\#L^* = p^{q_2r}-1$; it's also true that $K\cap L$ is a proper subfield of both, with $\#(K\cap L)^*=p^r-1$. (All these groups are cyclic.)
There are $\#F^*/\#L^* = (p^{q_1q_2r}-1)/(p^{q_2r}-1)$ distinct translates of $L^*$, which together partition all of $F^*$; in particular, all elements of $K^*$ are in some translate. Each time a translate intersects $K^*$, it does so in $\#(K\cap L)^*=p^r-1$ elements.
Therefore there are a total of $\#K^*/\#(K\cap L)^*=(p^{q_1r}-1)/(p^r-1)$ translates of $L^*$ that intersect $K^*$.