Minimal requirements for Kleene's recursion theorem
I stumbled upon a (partial) answer myself, in David Madore's essay about quines. I quote,


  A different, perhaps more satisfactory, way of stating the fixed-point theorem would be to eliminate the universality theorem from it, and to say: for every computable function $k$ there exists a $n$ such that $\phi_n(\dots) = k(n\dots)$. This corresponds more precisely to the intuitive content we have described. It is proved without the use of the universality theorem, using only the s-m-n theorem (for the actual proof, take the proof we have just given, and replace $\phi_{h(x)}(\dots)$ by $k(x\dots)$ everywhere). The advantage of formulating things like this is we see that it also works for primitive recursive functions (which satisfy s-m-n but not universality), so in effect a primitive recursive function can also make use of its own number. By applying the universality theorem (the function $\phi_{h(x)}(\dots)$ is computable, so we can call it $k(x\dots)$) we recover the fixed-point theorem as we have stated it. 


So Kleene's fixed point theorem goes through at least for primitive recursive functions. It might still be that there's an even simpler class of languages that it also applies to, and I won't accept my own answer in case someone else has more to say on that topic. But for now I'm happy knowing this.