Public policy PhDs do go into think tanks but they also go into academia i.e. at policy schools and in the policy lines at political science departments. Check the placement records of places like HKS. Some public policy PhD programs, like Michigan, are joint programs with political science giving you access to both job markets and training in a traditional discipline. If you really want to make the transition from engineering to public policy (where you are now) to political science, this might actually be your best option, as your application to the public policy program would be viewed more favorably, and if accepted by them, you would be accepted to both programs simultaneously.
You will still be hurt by a lack of letters from social scientists. Again, political science faculty simply cannot relate when faculty from other disciplines talk about how strong your work is. Other applicants (your competitors) will have letters from faculty *who the search committee knows personally or by reputation* which carry a great deal of weight. In public policy, the senior fellow letter might help.
Have you considered getting a US master's degree in public policy or political science, to build up your credibility and relationships with professors in this field?