One way to better understand the realistic chance of switching into Corporate Strategy or Corporate Development role is to do a job search for those roles and read through the job description. From my understanding ,most of the corporate strategy roles recruits from management consulting ranks, and corporate development is largely an M&A function requiring some prior connections to investment banking and/or consulting experience in due diligence.
While nothing is impossible, getting into either or the pre-req industry is difficult in mid-30s. Many of my classmates are at principal/associate partner level at MBB. That industry is really for candidates who are not married at the consultant level. In all honesty, the company will question your ability to work long hours and handle the travel requirement otherwise.
I would advise that you really understand those roles, and see if you would like those aspect of the role. Many of the corporate strategy roles have very little to do with strategy, and have very little influence on business direction. It's more of a reporting role. There's a lot of strategy involved in a Product Management role, especially at smaller companies. At smaller companies, PM role have a tremendous amount of influence on the direction of the company/products.