1. You should apply to LSE for its EME program since it seems like your mathematical preparation is good. That program does a far better at job at placing people at top PhD programs than any American masters.
2. You can still apply to predocs even after working. It is not uncommon for private-sector people to apply to these jobs. If your main weak point is research and letters of recommendations, then you may want to still apply for predocs.
3. Is there a reason why you are working first? It's not a bad thing, but I also want you to be certain that an economics PhD is actually what you want to pursue.