All valid points, ones which I have considered.
I'm teaching survival English to adults on a limited schedule so I necessarily have to simplify and excise some of the finer points of English grammar. The accent on my syllabus is to keep it lively and keep it moving, not to drill grammar forms endlessly.
I'm aware that some people use the "have you a...?" question form and I tell my students exactly what I wrote here: that it is strictly speaking correct but I know no one who uses it. This includes people from the US, Canada, Australia, England, Ireland, and Wales that I know or have known. I think I've met two people in my whole life who have used this form. Also, Italians in particular love this form because Italian public school language teaching is modeled on English from 50 years ago. I try to get them away from that. If they strongly prefer "have you a cat?" that's fine with me, as long as they know that they'll be in the minority. So, no offense to anyone who uses this form, but I tell my (adult) students my opinion on the matter and it's up to them whether to follow my advice or not.
I do make them aware of "have got" and I make extensive use of recordings so it appears repeatedly. They understand it, which is sufficient for me: teaching the proper use of have got vs. have, especially in the short answer forms ("yes I have got" crops up often) takes up too much time (I typically have 25 or so lessons per class), therefore I make sure they understand "have got" but are proficient in use with "have", since "have" has future, past, and other forms that "have got" lacks.
I incorporate audio from speakers from everywhere including BrE. Where feasible, I teach alternate ways of saying things. But I make no secret that I'm American and that my teaching will inevitably reflect this. It's a bit like not knowing anything about cooking and taking lessons from an Italian. What you'll be learning will be influenced by your Italian teacher and there are differences from French cooking (which some consider to be superior). Once you get to a certain degree of skill in knife handling, cooking techniques and so on you can decide to specialize in Italian or French or Spanish cuisine. But most of my students are in the "never really handled a knife" stage of their English careers so I teach them a simplified version of English based on what I know exhaustively rather than every single permutation, sort of like a one-week cooking course rather than a two-year culinary degree.
Finally, I and my students work for a US-based company so I have a solid reason to teach AmE.
Well, that last sentence is a matter of opinion. While you may not have met anyone who uses 'have you a lighter?', there are still quite a few of us around.
From what you have written, it appears that you do not expose your students to have got. That is probably the most common form in spoken BrE, hardly a minor dialect. Should they not be exposed to that, even if they don't use it? As Italians, they are, after all, closer to the UK than the USA. It could well be that "the majority of English speakers they are going to come into contact with" will be speakers of BrE.
Certainly for your students to use the auxiliary DO with the full verb HAVE is fine, though have you checked on whether there is a required form for your students to use? I think it would be unreasonable if such forms as 'I don't have' were to be penalised in an examination, but they could be.