An infinite decidable language that is a subset of $\overline { A_{TM}}$
Fix a special TM $M_d$ which infinitely loops on any input $x$. In other words, $M_d$ does not care about input $x$ and always infinitely loops on any input. Such a TM could consist of a single state and instructions such as $\delta(q_0, a) = (q_0,a)$ for all input symbols $a \in \Sigma$,  which tells $M$ to stay in the same state and never move the head. So, given its description $\langle M \rangle$ it is easy to recognize this machine by inspecting its states and instructions. 

Since $M_d$ never halts on any input $x$ the following language must be infinite

$$L = \{\langle M_d,x \rangle \mid x \in \Sigma^*, M_d \text{ is the fixed TM} \}$$. This language is also a subset of $\overline{A_{TM}}$ since $M_d$ never halts on any input $x$. Thus, if we are given $\langle M_i,x \rangle$ we simply decide by checking whether $i=d$, i.e., whether $M_i$ is the same as $M_d$ by inspecting $M_i$'s states and instructions meaning $L$ is decidable.