Taking BMAT without UKCAT is unusual, especially if you take the autumn BMAT exam. It is reasonably common to take UKCAT alone, but it means you cant apply to Oxbridge and some of the London medical schools. It used to be easier to get away with just BMAT but places like Bristol have changed their approach and now want a good UKCAT score. (DD crept in when they did not want either BMAT or UKCAT!).
Oxbridge offer a different approach. DD had patient contact in her first week. Oxbridge is regarded by some as more academic, and is six years rather than five, but some of the five year courses allow you to spend an extra year on an intercalation in your chosen speciality, not necessarily at the same medical school.
It is worth really thinking about what you want. Different does not mean better.
I don't get the 'only take three A levels'. If you are likely to get A* predictions and can manage 4, why not. DD took 5, played sport at high level, held a senior leadership role at her school, and did plenty of shadowing and volunteering. She was also up to date with Grays Anatomy and had time to hang out with friends. Medics are often busy people who are good at juggling. It does not necessarily help with getting in to med school, though may have been an advantage at BMAT. (DD decided that she did not want to go, and though many of her friends did, she has no regrets about branching away from the Oxbridge/London triangle.) The big advantage for her is that it gives her more intercalation options and would have given her a broader ranger of options had she changed her mind about medicine. Maths proved a big advantage in her first year.
In terms of choices it might be worth one aspirational, two achievable and one fall back. Each medical school selects in different ways, so strong GCSEs, strong A level predictions, and UKCAT and BMAT scores all come into play. DDs school normally advised against applying to both Oxbridge and Bristol as they were both competitive albeit in different ways. (Bristol's applicant to place ratio reached 17 to 1 before they started using UKCAT, in part becausethey were generous with contextual offers.) In contrast Queens Belfast has a good reputation but low applicant to place ratio, and probably lower still for students from England, so for students with the right profiles can be a fall back.
My advice is to start adding the different data to a spreadsheet so that your DC can start crossing off places that dont appeal or places who are looking for students with different profiles.
On MN it can feel as if everyone gets 4 offers. This really is not the case, even for students with top flight academics. It is worth treating it as a two year process, not least to help keep the pressure off in Yr 13.