I think getting sponsored would require sharper targeting of the role you could do. I would definitely suggest going the sponsored transfer route, because it would allow you to get "on the board" with a job and start building a career, and a case, for transferring you to a similar position overseas.
ONE CAVEAT though - when you interview for a position at a company that has large, multinational presence, make sure you ask a question about global opportunities and how the company enables workers to access those opportunities. I took a job 4 years ago with a company with a very large presence in the UK, with the intention of learning and then transferring. From my first day there I indicated my intentions of taking an overseas posting, there was a yearly role dedicated to my department that allowed someone to rotate into the UK for a year, etc. What wasn't indicated in the interview was that you had to do some heavy politicking and networking to even have a shot at an overseas role, and the rotational role was pulled after a year because of poor company growth. The business units were so divided and siloed, in addition to a lack of a good international transferal practice, that even applying internally for an overseas role had the same odds as applying externally. I finally gave up, quit (it was a highly toxic environment anyway), and moved anyway (but I have work rights for the UK).
I would suggest you check out E&Y - they are doing a heavy recruitment drive/expansion at the moment and do have a consulting practice or any number of slots someone with some sort of strategic background could fit into. I just interviewed with them in the UK and while I had to prove to the recruiter I could work, E&Y honestly weren't bothered and would have sponsored, so long as London was where I really wanted to be.