$\newcommand\S{\mathcal S}\newcommand\R{\mathbb R}$For $x\in(0,1)$, let $g(x):=\frac1x$, and then let
$$f(x):=e^{-g(x)-g(1-x)}$$
for $x\in(0,1)$, with $f(x):=0$ for real $x\notin(0,1)$.
Then $f\in\S_0^3(\R)$ -- see the details on this at the end of the answer.
As noted in Willie Wong's comment, you can rescale $f$ vertically and horizontally to get desired $C$ and $A$ (and you said you do not need $B$ to be small).
Details: Everywhere here, $x\in(0,1)$. We have
$$f'(x)=P_1(x)f(x),$$
where $P_1(x):=\frac1{x^2}-\frac1{(1-x)^2}$. It is now easy to see by induction that for any $r=0,1,\dots$
$$f^{(r)}(x)=P_r(x)f(x),$$
where $P_r(x)$ is a polynomial in $\frac1x\,,\frac1{1-x}$ of total degree $\le2r$, containing $\le C^r$ monomials with coefficients $\le C^r r!$ in absolute value; here and elsewhere $C$ denotes various universal positive real constants. Also, for any nonnegative integers $m$ and $p$ such that $m+p\le2r\le2q$ we have
$$\dfrac1{x^m(1-x)^p}\,e^{-g(x)-g(1-x)}\le e^{-m-p}m^m p^p
\le q^{m+p}\le q^{2q}.$$
Thus,
$$|f^{(q)}(x)|\le C^q q^{3q}.$$
By a more accurate accounting on the coefficients of the monomials of $P_r(x)$, one can get
\begin{equation*}
|f^{(q)}(x)|\le C^q q^{2q} \tag{10}\label{10}
\end{equation*}
and thus $f\in\S_0^2(\R)$.
Indeed, for $r=0,1,\dots$,
\begin{equation*}
f^{(r+1)}(x)=(P_r(x)f(x))'=(P'_r(x)+P_r(x)(\tfrac1{x^2}-\tfrac1{(1-x)^2}))f(x),
\end{equation*}
so that
\begin{equation*}
P_{r+1}(x)=P'_r(x)+P_r(x)(\tfrac1{x^2}-\tfrac1{(1-x)^2}).
\end{equation*}
Consider any monomial $m(x)=c(\frac1x)^i(\frac1{1-x})^{p-i}$ of the polynomial $P_r(x)$
in $\frac1x\,,\frac1{1-x}$; this monomial is of total degree $p$ and with coefficient $c$. Then $(m(x)f(x))'/f(x)$ is the sum of four monomials of $P_{r+1}(x)$. Two of these four monomials are monomials of $P'_r(x)$ (say they are of the 1st kind) and the other two monomials are monomials of $P_r(x)(\tfrac1{x^2}-\tfrac1{(1-x)^2})$ (say they are of the 2nd kind).
The total degree of each of the two monomials of the 1st kind is $p+1$ and the absolute value of its coefficients is $\le p|c|\le2r|c|$. The total degree of each of the two monomials of the 2nd kind is $p+2$ and the absolute value of its coefficient is $\le |c|$.
So, if a monomial of $P_q(x)$ is obtained by $k$ steps of the 1st kind (and, hence, $q-k$ of the 2nd kind), then it is of the form
\begin{equation*}
M(x)=c(\tfrac1x)^i(\tfrac1{1-x})^{p-i}
\end{equation*}
with
\begin{equation*}
p=k+2(q-k)\quad\text{and}\quad |c|\le (2q)^k\le2^q q^k.
\end{equation*}
Recalling now the inequality $u^s e^{-u}\le s^s$ for $u>0$ and $s\ge0$ (with $0^0:=1$), we see that
\begin{equation*}
|M(x)f(x)|\le2^q q^k i^i (p-i)^{p-i}\le2^q q^k p^p=2^q h(k),
\end{equation*}
where $h(k):=q^k (k+2(q-k))^{k+2(q-k)}$. Note that $h$ is log convex, $h(0)=(2q)^{2q}$, and $h(q)=q^{2q}$. So, $h(k)\le(2q)^{2q}$ for all $k\in[0,q]$.
So, for any monomial $M(x)$ of $P_q(x)$ we have
\begin{equation*}
|M(x)f(x)|\le2^q (2q)^{2q}=8^q q^{2q}.
\end{equation*}
Since $P_q(x)$ contains $\le4^q$ monomials, we conclude that \eqref{10} holds. $\quad\Box$
$\newcommand\al\alpha\newcommand\be\beta\newcommand\R{\mathbb R}$This is not true in general. E.g., suppose that $m=1$, $\mu=N(0,1)$, and $r_n(x)=r:=1/n$ for all $n$ and $x$. Let $f$ be the standard normal pdf.
Then for each real $x$ (and $n\to\infty$) we have
$$\al_n(x)\sim\int_{x-r}^{x+r}(y-x)^2\,dy\,f(x)=\frac{2r^3}3\,f(x).$$
Next, letting $X$, $Y$, and $Z$ denote independent standard normal random variables, we get
$$A_n=\int\al_n f=E(X-Y)^2\,1(|X-Y|\le r)
=2EZ^2\,1(|Z|\le r/\sqrt2) \\
\sim 2f(0)\int_{-r/\sqrt2}^{r/\sqrt2}dz\,z^2=f(0)\frac{r^3\sqrt2}3.$$
So, for $F:=f$ and
$$L_n:=\frac1{A_n}\int F\al_n \,d\mu$$
we have
$$\liminf_n L_n\ge\int f\frac{\sqrt2}{f(0)}\,f \,f=L:=\frac{2}{\sqrt{3 \pi }},$$
whereas
$$R:=\int F\,d\mu=\int f^2=\frac{1}{2 \sqrt{\pi }}<L.$$
So, $L_n\not\to R$. $\quad\Box$
Best Answer
Start from the estimate (with $q = K$ and $k = K+1$) (valid for $x > 0$) $$ |f^{(K)}(x)| \leq C B^{K} (K+1)^{1/8} K^{K/2} x^{-K-1}$$ Integrate back from infinity $K$ times, you get $$ |f(x)| \leq C \frac{B^K (K+1)^{1/8} K^{K/2}}{K!} \frac{1}{x} $$ Using Stirling's approximation $$ |f(x)| \lesssim \frac{e^K B^K (K+1)^{1/8} K^{K/2}}{K^{1/2} K^K} \frac{1}{x} $$ with the implicit constant independent of $K$.
Take the limit $K \to \infty$ you see that the RHS converges to 0. This shows that the only function with the specified property is $f \equiv 0$.