KC,
What you need to maximise your chances is legal work experience, however you can find it. I wouldn't recommend doing an LLM unless you want to do it purely for your own academic interest, it won't make any difference to applying for training contracts - particularly as you already tick the degree level academics box already. From my perspective - it's an expensive waste of a year when you could be doing something much more useful to get that TC.
How many TC applications are you making/ have you made, and are you getting through to interview stage or not?
Re work experience this is what I would do (and mostly did):
1) relentlessly apply for vac schemes at firms that offer them - it's a number games, the application need to be good quality, but you also need to do as many as you can
2) continue applying for paralegaling/legal assistant/company secretarial roles - again as many applications as you can.
3) - do you have any contacts through friends or family who work in companies that have a legal team/function/solicitor and can arrange some work experience, or a paralegaling role for you that way? I had no background or contacts in law but managed to gain some incredibly useful work experience from family friends who wern't lawyers, but worked in an office where they could ask the legal team "can a friend come in for a week of work experience" - this then led to a summer working as a legal assistant for the legal team of a local company.
I wouldn't pin your hopes on wanting to work for a specific firm and qualifying in a specific department, particularly a relatively niche area like International Arb & the firms that offer is (Like W&C, HSF etc... tend to be those wanting top top academics. All big firms have talented individuals working for them and who the head of private equity is is frankly irrelevant to a trainee or NQ - You'll be working for associates or senior associates and how they treat you is far more important. Culture is important and it does differ between firms, but ultimately if you're looking to do international work at city firms the culture is going to be pretty hard nosed and involve regular long, stressful hours with demanding clients wanting another all-nighter pulled to get something done. Not to say training at a city firm isn't great experience (it is), but it has its costs and drawbacks too. Obviously when you're applying you need to find some hook to answer the "Why Firm X" question but the reality is as a trainee or an NQ it's much of muchness which city firm you work for within their broad categories.
Realistically - do you have a reasonably chance of training at a top city or US firm that offers Int Arb as a seat? Unfortunately I think the answer here is no. I have a former international arbitrator in my team and he has a Oxbridge first and top A levels, very sharp, very astute
What you do have a chance of is finding a training contract *somewhere* and that needs to be your mental approach. That means working on increasing the work experience mentioned above, and applying for as many training contracts as you can in London, or any other area you would like to work in that does commercial work, many of these firms will be smaller that what you might want, but unless you end up with multiple offers you can't be fussy. Apply to the big firms too, because you never know..., but apply, apply, apply.
It's really hard to put chances into numerical terms because we're all different, your CV might catch someones eye, an interview goes incredibly well, you only need 1 lucky break etc... There's a chance, not a good chance, not a remote chance, but a chance. I would give it your best shot for up to 2 years so you've had at least 2 rounds of application cycles to see how it goes before re-appraising whether you want to keep trying.