Why bother with by-elections for a toothless body? If the voters (or the parties) pick people who die in office, tough luck. If the Lords would actually have some power then I could see the need for by-elections, but as it is, there is no need, or at least none that could not be put off until the next nationwide vote.
The draft bill and the joint committee report propose filling vacancies with substitute members until the next election (unless it's close to the next election - the committee recommends 1 year - in which case it would be left vacant). For some reason they propose filling the vacancy with the first unsuccessful candidate for the party at the previous election, which could involve going back to election results that may be more than 10 years old.