AFAIK, there is std::tr2::dynamic_bitset in (and I believe, only in) GCC (reference link here). I think it is to replace std::vector<bool> in a near future, but it is not yet in the standards. It is at least in a "usable" state if I remember correctly.

There's a simple way to use static bitsets (with minimal overhead) when dynamic bitsets are needed.

template <uint32_t N> // size_t might give stack frame related warnings
void helper(int n) {
    if (n <= N) {
        bitset<N> b;
        std::cout << "N = " << b.size() << '\n'; // implementation goes here
    } else {
        helper<2 * N>(n);
    }
}

void solve(int n) {
    helper<1>(n);
}

And if you want to limit $$$N$$$ to a certain limit, you can do it using std::enable_if (now you can even use size_t without warnings):

static constexpr size_t MAX_BITSET_SIZE = 10000;

template <size_t N>
std::enable_if<(N <= MAX_BITSET_SIZE)>::type helper(int n) {
    if (n <= N) {
        std::bitset<N> b;
        std::cout << "N = " << b.size() << '\n'; // implementation goes here
    } else {
        if constexpr (2 * N <= MAX_BITSET_SIZE) {
            helper<2 * N>(n);
        }
    }
}

void solve(int n) {
  helper<2>(n);
}

And if you want to be more space efficient about it, consider replacing 2 * N by static_cast<uint32_t>(1.5 * N + 1) or something similar everywhere, though the smaller the increase, the more functions and if checks there will be. In the worst case, if the increase is linear, you'll either have a bloated binary (with potential instruction cache misses) or the compiler will fail to compile the code due to out of memory errors, not to mention the linear search time in this recursive definition.