For the case $t=2$, letting $A_1:=A$, $A_2:=-B$, $A_3:=[-k,k]$, and $A_4:=[-l,l]$, the sum in the left-hand side is $$ \sum_{a_3\in A_3, a_4\in A_4} |A_1\cap(-A_2+a_4-a_3)| $$ which, taking into account the symmetry of $A_4$, is the total number of representations of $0$ in the form $0=a_1+a_2+a_3+a_4$ with $(a_1,a_2,a_3,a_4)\in A_1\times A_2\times A_3\times A_4$. It is known that this number of representations is maximized, over all quadruples of sets $(A_1,A_2,A_3,A_4)$ of prescribed odd cardinality, when each set $A_i$ is a block of consecutive integers centered at $0$. This is a partular case of the rearrangement inequalities due to Gabriel, Hardy, and Littlewood; see this paper (particularly Theorem 1) for generalizations, references, and the historical background.

This answers your question in the case where $t=2$. For $t>2$, the inequality you are asking about may well be unknown, and yet it should be possible to prove it using the same approach as in the case $t=2$ (basically, by a smart induction, removing simultaneously elements from the sets $A_i$).

For the case $t=2$, letting $A_1:=A$, $A_2:=-B$, $A_3:=[-k,k]$, and $A_4:=[-l,l]$, the sum in the left-hand side counts the number of quadruples $(a_1,a_2,a_3,a_4)\in A_1\times A_2\times A_3\times A_4$ with $a_1+a_2+a_3+a_4=0$. It is known that this number is maximized, over all quadruples $(A_1,A_2,A_3,A_4)$ of sets of prescribed odd cardinality, when each set $A_i$ is a block of consecutive integers centered at $0$. This is a partular case of the rearrangement inequalities due to Gabriel, Hardy, and Littlewood; see this paper (particularly Theorem 1) for generalizations, references, and the historical background.

This answers your question in the case where $t=2$. For $t>2$, the inequality you are asking about may well be unknown, and yet it should be possible to prove it using the same approach as in the case $t=2$ (basically, by a smart induction, removing simultaneously elements from the sets $A_i$).