How to generate a binary string so to minimize weighted bitwise distance?
It is not possible to uniquely determine $i$-th combination because a function $f: \{0,1\}^* \to \mathbb{R}$ assigning distance to a string is not generally injective., that is - there may be multiple strings that correspond to a specific distance $d \in \mathbb{R}$ (which you example actually presents).

Also, it is not possible to know in advance all the possible distances as to determine which strings could be in $i$-th position, without actually calculating all the distances. It would be like knowing the shortest route to some destination by examining just that route.

I would suggest a different approach (but it will not make you happy as I generally think there is no way to achieve this in its fullest without generating all combinations); creating an algorithm that takes as input not $i$, but the distance $d$ and returns strings that are $d$ distance from your base string (note that it is in fact irrelevant).
You can then eliminate from further process the weights that are larger than $d$ (and even ones equal to $d$, with further care).


If you want all strings $d$ distance away and remaining weights vector is not too long, you can generate all subsets (a power set) of weights array and checking which has the sum $d$, but considering your restrictions on the solution - you do not want that because it would generate all the binary strings in the process.
If you need just one string on distance $d$, no matter which, you can implement some backtracking search/sum on the weights array to find a combination that sums to $d$. (You can then store ones you got and repeat this process ignoring ones you found already, if you still want all.)
You can model your problem as finding a path of length $d$ in a fully connected directed graph where you graph has $N$ nodes and the weight of an edge going to node $i$ is $weights[i]$. Then you can maybe use some optimized graph path finding (heuristic) algorithms, but that seems like an overkill.


I am quite sure this problem is in NP.