Subset of a regular language, for each of whose words there exists an element with the same number of 1s in the other regular language
The idea is as follows:


We run two automata in parallel: one verifies that the input is in $A$, and the other verifies the other condition.
The second automaton simulates an automaton for $B$, in the following way. At any point in time it may simulate reading a $0$. It ignores all $0$s in the real input, and forwards all $1$s in the real input.


I'll let you construct an NFA given this description.