Here is a second attempt (see edit history for previous version).

For each $t\in\mathbb{N}$, let 
$$P_{i,j,k,t}=\{1_{i,j,k,t},\ldots,n_{i,j,k,t},\ldots,\gamma(i,j,k)_{i,j,k,t}\}$$
 (so that for each choice of $i\in I$, $j\in J$, $k\in K$, and $t\in\mathbb{N}$, we have a disjoint set of size $\gamma(i,j,k)$). 

For each $t\in\mathbb{N}$, let 
$$Q_t=\{a_{k,t}\mid k\in K\}$$ 
(so for each $t\in\mathbb{N}$, this is just a copy of $K$, up to relabeling). 


Let 
$$X=\coprod_{t\in\mathbb{N}}\left(Q_t\coprod_{\substack{i\in I,j\in J\\\k\in K}}P_{i,j,k,t}\right).$$
Define 
$$\Omega_j=\coprod_{i\in I,k\in K}P_{i,j,k,1}\subset X,$$
and $f_i:X\rightarrow X$ by 
$$f_{i_0}(n_{i,j,k,t})=\begin{cases}a_{k,1}\text{ if }i=i_0,t=1\\\ n_{i,j,k,t+1}\text{ otherwise}\end{cases}$$
$$f_i(a_{k,t})=a_{k,t+1}$$

Thus 
$$f_{i}^{-1}(n_{i,j,k,t})=\begin{cases}\emptyset\text{ if }t=1,2\\\ \{n_{i,j,k,t-1}\}\text{ if }t>2\end{cases}$$ 
$$f_i^{-1}(a_{k,t})=\begin{cases}\coprod_{j\in J}P_{i,j,k,1}\text{ if }t=1\\\ \{a_{k,t-1}\}\text{ if }t>1\end{cases}$$ 
We choose $p_k=a_{k,1}$.

Thus $f_i^{-1}(p_k)\cap \Omega_j=P_{i,j,k,1}$, so $|f_i^{-1}(p_k)\cap\Omega_j|=\gamma(i,j,k)$.

Unfortunately this still doesn't address your size concerns, i.e. the preimage of any element of $X$ being countable, because if $J$ is uncountable then $f_i^{-1}(a_{k,1})$ is uncountable (I added the whole mess with the $t$'s to make the preimages of all the other elements countable). I'll leave this as a community wiki, and if anyone sees a way of fixing it they are welcome to edit this.