$N+1$ button presses are required just to confirm a length-$N$ name, regardless of its letters (an A button press upon selecting each letter, plus a final + button press at the end). Therefore, if $K<N+1$, then no valid name exists. Otherwise, let $B=K-(N+1)$ be the number of additional </> button presses required.

Let's consider how many such button presses each letter in a name contributes. Let $m(p,x)$ be the number of button presses required to move the cursor to a letter $x$ from the previous letter $p$ (with the previous letter before the first one assumed to be A). We'll either repeatedly press < or >, whichever requires fewer presses. These two options require $|x-p|$ and $26-|x-p|$ button presses (in some order), giving us $m(p,x)=min(|x-p|,26-|x-p|)$. We can observe that, given any previous letter $p$, it's possible to choose some next letter $x$ such that $m(p,x)$ has any wanted value between $0$ and $13$, inclusive.

We can then observe that a valid name exists if and only if $B\le 13N$. Furthermore, we can construct a name for a given $B$ value by greedily choosing letters one by one such that $m(p,x)$ is maximized each time (up to at most $13$), but without exceeding the remaining allotment of required button presses.