Masters to break into IB from medical school (no experience)

Hi, 

I’m looking for any advice for my situation. 
 

I am a final year medical undergrad at a  London university, red brick non target. I’ve also got a 1 year first class honours management related degree (intercalated BSc) at a target business school in London done midway through medical school (very common to do). Modules like accounting, healthcare economics, worked with Big4 in dissertation project, finishing v high in year. Firsts in all and overall. 

I have become interested in IB incredibly late and as a result have zero internships.

My plan which I would appreciate for someone to revise/destroy/ laugh at is as follows: 

1) crush the gmat and apply to do a masters at LBS/LSE/HEC in management (MiM). Do the CFA exam before I start to show a genuine interest. Do a 16 month masters/ 2 years to increase  the chances of an internship. In the mean time get some experience helping med start ups in finance way (anything finance related). Even if this doesn’t get me into IB straight away it might get a finance job at Big4 which I could try and utilise later or via an MBA?
 

2) the management masters at those  universities because of their name and also because at the moment I don’t qualify for a Masters in finance. I’ve seen a lot of LBS MiM in IB now so it’s not unheard of, not so much LSE MiM. 
 

3) alternatively try for consulting work in (healthcare sector) that does some finance work and then with that experience apply to LBS MFa 16 month programme or the MiM. Would a year working harm my chances though? Quite a lot of medics move into consulting and healthcare consulting is a natural fit. 

4)I have zero chance and am better off doing an MBA in five years time ?

Would it be worth applying to any other universities/courses. I don’t care where they are as long as they could help. 
 

5) Is the Healthcare banking route the most sensible? It seems logical as I do know of a few doctors/ med students who went into IB (last minute or as a career change) and all did so via the healthcare investment banking route which is what my number one choice would be.

Some did so via CFA and cold calling to an off cycle they converted in their final year with no prior experience and from a non target, another via an MBA.

I know I’m not getting in without a masters/mba , I am well prepared to follow through this.