I think that your worries would be best addressed by speaking with an economics professor from your undergrad with whom you took a PhD level class. Send them your CV and a brief discussion of why you're interested in doing an Econ PhD after working in finance as well as your concerns about your application profile. They should be able to provide decent guidance about whether you even need to take a masters/predoc or not. However, I do think that you're vastly over-weighting the small "sub-optimal" parts of your profile and vastly under-weighting the good parts of your profile.
With respect to EME, many people take the EME course after undergrad, so they can't have had that substantial amount of research experience yet (compared to a predoc), and every year, the top EME students get placed in fantastic programs. Many EME students also do not major in econ in undergrad. I don't see why your application results would be so different from the profile of successful EME students given these facts about the program. Your primary problem is getting good letters of rec; getting top grades in EME would give you the necessary letters from LSE professors (where else are EME students getting their letters, after all?) In contrast, the track record of terminal American masters is just much worse than the track record of the EME program. No matter how strong a candidate you might be, don't forget that there are also many other candidates who are just as strong. Your best bet for getting into good programs is to follow the path of other successful candidates. For candidate who don't have good recs, the best bet is applying to a predoc or to a non-American masters like EME.
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