> Only 92 of the 842 peers are hereditary currently, so it’s not really necessary to convince them to agree;

As I understand it, it was necessary (in order to pass the bill without the delay the Lords can impose) to secure a deal on the hereditary peers (not with them), because the Conservatives (the largest Lords faction) and many of the cross-benchers among the life peers, a sufficient number in total to delay the bill (the Lords can't actually block it permanently) oppose the bill, not just a group among the existing hereditary peers.