Implementing random.chose with urandom in python

I was playing with urandom based implementation of random.choice and I'm wondering what people here would think about where I got and where I'm going. First some context.

My first way to utilize urandom to make a uniformly random selection is to

1. Calculate the number of bits required to represent the indices of the collection.
2. Generate that number of random bits until the value represented doesn't exceed the size of the collection.

Which is what I do in the following code:

import math
import os

def urandSelect(chrs):
    bits = math.ceil(math.log2(len(chrs)))
    idx = len(chrs)
    while idx>=len(chrs):
        idx = int("".join([bin(b).split("b")[1] for b in os.urandom(math.ceil(bits/8))][-bits:]),2)
    return chrs[idx]

Now here's the rub; worst case runtime. Hypothetically this could generate a random idx>=len(chrs) any number of times before actually returning. So while the expected runtime is reasonable the worst case runtime is infinite.

So my question is this: how would you go about making a uniform random selection with a bound worst case runtime using urandom as your RNG?

A side note, the idea of just generating a large number of random bytes and dividing down into the range required to index the collection has been suggested but the non-uniformity arising from 2^nbits % len(chrs) != 0 is a deal breaker.