Giving her more time is simply going to mean giving her more time in which to keep doing the same thing. You have to shake things up if you want things to change. If your boyfriend passes 100%, his mother
does look crazy to other people every time she does this. Does she realise that? Have you pointed this out to her? Rather than focusing on how it makes Lucas feel, why not explain to her that she's making herself look like she's lost her marbles?
I had a couple of relatives who would do this to me constantly, despite my regular explanations of how it made me feel and how they were putting my safety in danger by outing me. They kept doing it regardless, so after many months of patiently explaining where they were going wrong I finally gave them an ultimatum: treat me with respect, use the correct name & pronouns, and do not talk about my confidential medical history without my express permission. I told them that if they chose to continue humiliating me in public, I would act like they're crazy and I'd humiliate
them.
They didn't heed the warning.
So the next time they misgendered me in public I used my acting background to act like they're a mental patient on day release. I would get all embarrassed & apologise to the third party for my relative's obvious illness. I'd say something like "I'm so sorry, my relative is on day release from the hospital. It's so sad how she's lost touch with reality since The Incident and she's overdue for her meds. I was hoping we could stay out longer but it looks like it's time to get her back to the hospital now." The third party would leave very quickly after that; nobody wants to be part of such an awkward conversation!
My relatives would be livid about this of course, but then as soon as the third party was out of earshot I'd turn to them & say that this is exactly what they can expect if they don't respect my identity and my privacy. And that I would do it
every time they chose to overstep the mark. Why should I be the one feeling humiliated due to someone else's misbehaviour? I can quite happily spread some humiliation around if they want to start something!

I only had to do this twice before they learned to respect my privacy!

I also never answer to the wrong name & pronouns (I immediately blank you & completely ignore anything you're saying if you use the wrong ones). The only way to stop her from doing it is to give her consequences for her behaviour... and then stick to them.