The move from ops mgt ( hardcore ops such as 6 sigma, etc) to strategic consulting ( and that too PE) aint gonna be easy!! Really have to show skill transferrability. How can your ops mgt skills be useful in a strategic capacity in PE? This is what you have to come up with and sell this like crazy.
Youe easiest route may be to just do an MBA , focus on strategy and finance in school and try to get a jon in PE strategy upon graduation.
Note that PE is notoriously tough to get into ...its an old boys network and they rarely advertize jobs, even to recruiters. They fill positions based on word of mouth, and who knows who...try and get into this circle!!You gotta know the right ppl.
Also note that to get into strategy in PE, you almost definately need to goto a top business school. If you dont, your chances are severely limited , unless of course, you know ppl, very influential ppl.