Is there a formulation of Rice's Theorem that does not involve admissible (or Gödel) numberings?
tl;dr:

(1) While (a) there is a kind of Rice's Theorem for your example construction, (b) it seems to me unlikely that one holds for realizable numberings in general.

(2) Your particular construction of a code $\psi$ with a fixed encoding for some $f$ has lots of (provable) deficiences, and I think other numberings can only get worse from there, so I don't think it's a very good programming environment, though I suppose that's a matter of what you're willing to trade off in order to have that fixed encoding.

Elaboration:

(1a) Rice-type theorem for your example construction. Let $\psi$ be the encoding from the comments, namely $\psi_0$ is the all-zeros function, and $\psi_n = \varphi_b + \delta_a$ where $n = (a,b)$ under some fixed computable bijection $\mathbb{N} - \{0\} \to \mathbb{N}^2$, $\varphi$ is a standard (acceptable) numbering, and $\delta_a$ is the Kronecker delta function, which is $\delta_a(a) = 1$ and $\delta_a(a') = 0$ for all $a' \neq a$. Let $P^{(1)}_1$ denote the subset of $P^{(1)}$ consisting of functions that have at least one nonzero output value (note that these are precisely the functions $\psi_{(a,b)}$ such that $\varphi_b(a)$ halts). Then for any subset $\emptyset \neq S \subsetneq P^{(1)}_1$, the function 
$$s(a,b)=1 \text{ iff } \psi_{(a,b)} \in S$$
is not computable. We mimic the proof of Rice's Theorem; suppose $s$ were computable. Let $(a_0,b_0)$ be such that $\psi_{(a_0,b_0)} \notin S$ and let $(a_1,b_1)$ be such that $\psi_{(a_1,b_1)} \in S$. Let $D(p)$ be the total computable function which proceeds as follows: 


$D(0)=0$
$D(p)(n)$ = if $s(p)=1$ then $\psi_{(a_0,b_0)}(n)$, otherwise $\psi_{(a_1,b_1)}(n)$


Now, if $s$ is computable, then so is $D$, and furthermore, $D$ is in $P^{(2)}_1$, so there is some index $(a_d,b_d)$ for $D$. Then consider $D((a_d,b_d))$: if $s((a_d,b_d))=1$ then $D((a_d,b_d)) = \psi_{(a_0,b_0)} \notin S$, and if $s((a_d,b_d))=0$ then $D((a_d,b_d)) = \psi_{(a_1,b_1)} \in S$, so either way $s$ gets it wrong.

(1b) The above relied on the fact that although we don't have a total computable compiler from $\varphi$ to $\psi$, we at least had a partial one which covered a nontrivial set of functions. I wouldn't be surprised if you can cook up a realizable numbering where there's essentially no such partial compiler.

(2) Since a $\psi$-Rice's Theorem fails for $\{f\}$, $\psi$ also must not satisfy the S-m-n Theorem (currying). In particular, there is some $n$ such that $\psi_n(x,y) = \varphi_x(y)$ (i.e., a universal TM). But then there is no computable program transformation $t$ such that $\psi_{t(m,x)}(y) = \psi_m(x,y)$ for all $m,x,y$, for if there were, $t(n,x)$ would enable you to decide whether $\varphi_x = f$, which is undecidable by the usual Rice's Theorem. Currying is a pretty basic feature for a programming environment to lack. I think one can prove a similar thing not just for currying, but for infinitely many program transformations, probably whenever the transformation is not finitely different from the identity map and has $f$ in its image.