My simple thoughts...The problem with mental health checks, is flawed at the core dependency. Like depending on criminals to be honest and adhere to laws, you are expecting someone with mental health problems to have clarity about their mental capacity. Example - those who rely on meds, in some stages, they are the ones who either forget to take their meds because they are really suffering, or they start feeling better (and don't think they need them any longer). These have been long standing concerns and problems with those in the jail system.
So how can you rely on a mentally unhealthy person, to know how to speak for their self (or how to answer their doctor)? It would definitely depend on the type of mental health problem. I think someone with bi polar tendencies, or split personalities, will answer you differently on different days, for example.
People who sell weapons, are like bartenders - they know nothing about you when you walk up, how are they even qualified to make a judgement in the 10 seconds you ask to see or make a purchase? I've never understood that. I've talked to many a Houston defense lawyer who's been on both sides of that debacle.
I think in short - regardless of whether they are a criminal, or have a mental health problem, if they want a weapon, they WILL get one. They might try to buy one, and can appear "normal" and will be successful. In the rare chance they are honest, they won't just stop once and say "on well", but they will try alternate means to get a weapon. We have to remember - just because they were honest with a doctor or retail Wall-World clerk, and are denied, isn't the end of their quest.