A bitset with 1,000 bits fits into two cache lines (64 bytes per cacheline = 512 bits per cacheline), and the very first iteration will shrink this to one cache line (it halved the search area), meaning AT WORST this causes one cache miss.
Wait a second, you're mixing a bunch of things: First you mentioned binary search over a vector, not a bitset. Second, you're suggesting binary search for iteration? That doesn't makes much sense, it wouldnt help you for iterating over a bitset.
For binary searching through a vector (what you initially suggested) you'd need 1000 ints for the IDs, thats 4kB of data and it'd really involve a bunch of cache misses.
There are plenty of ways to iterate over the bits of an int, but it has to simply check if each value of the backing array has any single bit on before fetching the next one, regardless of the technique you use to iterate over the bits themselves in the single int.