HAP landlord here -
I’d tell them, they may already be a HAP LL.
the biggest issue most LLs have with HAP is the knowledge gap of what it actually entails - it’s quite a long and VERY probing process (3.5 months in my case from application stamped received by council to having rent in my account - April 10th - Jul 28th). That wasn’t unduly slow either, it’s taking 8+ weeks to process claims, and then they missed the June payment run and had to wait to next month.
The risk the LL has is that if you stop paying the council, the council immediately cut the landlord payments loose.
People refusing HAP aren’t necessarily all discriminating against you either - there’s a large list of banking, tax and LTP proofs required - so if someone isn’t fully (and I mean FULLY) legit, they can’t touch HAP even if they wanted to.