I think it is interesting because you will have the technical skills to do strategy work, but many strategists like to think of themselves as "economists" (quotes there for a reason ...), so I can imagine some committees giving you more intense scrutiny despite in reality you likely being far ahead of most people starting Strategy PhDs. But I imagine, for the schools you want, you are going to have to get that GRE way up. It might be better to take the GMAT instead and see if you can get 700+ there.
You should take some time to read articles published in Strategic Management Journal (the most "econ" of all strategy journals IMO), and get an idea for what people are working on.
If your school is top 50 for strategy (I can't make assumptions based on econ department), transferring might be a good step.