A question that may help you reflect on this - have you seen many roles like this being advertised, or do you know of many people who've made such a move? Historically these strategy firms have recruited straight from college / university and then the MBA entry level is the main next entry point. Experienced hires are made where the firm has a major surge in client demand (or loss of consulting staff) in a particular specialist area - and so brings people into the business at other levels to address this. However, that's usually at more senior levels where you'd be expected to have a specialist area. At more junior levels you'd be a generalist and I would say bringing in "experienced hires" is much less common. Consultancies just vary the size of their next graduate intake to address shortages at the junior end of the firm.
I hope that helps - I'm not saying it's impossible to make such a move but I would say it's uncommon. The MBA route would be a far more proven way of making this switch (albeit you'd need to get accepted onto a top course or these strategy firms would be unlikely to take you).