Key-Bijective Secure Symmetric Encryption Methods In AES, an N-bit message is mapped 1-to-1 message-to-output (for a fixed key).
If we use a key with N bits, however, the key is not mapped 1-to-1 key-to-output
(for a fixed message).

So, is there any secure symmetric encryption method with a 1-to-1 key-to-output
mapping?

In the end, I'm just looking for a hash function h(m,k) with no collisions
over m (for a fixed k) and no collisions over k (for a fixed m), but it seems
easiest to leverage encryption methods research.
You are asking for a function $h(m, k)$ such that for fixed $m$, $k \mapsto h(m, k)$ is a permutation, and for fixed $k$, $m \mapsto h(m, k)$ is a permutation.  Obviously $m$, $k$, and the output of $h$ must be in the same space.  Suppose they live in a group $G$ (written additively), e.g. they're bit strings interpreted in $\operatorname{GF}(2^n)$.  For any fixed permutations $\pi$ and $\sigma$, $(m, k) \mapsto \pi(m) + \sigma(k)$ satisfies this.  (It seems likely that this is the only shape $h$ can have, but I haven't ruled others out.)

However, you presumably have some security goals in mind.  For example, you probably want to require that $\pi$ and $\sigma$ be difficult to invert; otherwise, a known-plaintext attack would lead to key recovery from the ciphertext $c = h(m, k)$ by $(m, c) \mapsto \sigma^{-1}(c - \pi(m))$.  One option would be $x \mapsto (g^x - 1) \bmod p$ where $p$ is a large prime and $g$ is a generator of $(\mathbb Z/p\mathbb Z)^\times$; this is a permutation of $\{0, 1, 2, \dots, p - 2\}$, inverting which is tantamount to computing discrete logs.  (One could also use $x \mapsto x^3 \bmod pq$ for large secret primes $p$ and $q$, if one wanted a permutation with a back door.)

It might help to know what you wanted to use this for!