Is there a route to portfolio manager with CS undergrad?

I'd like to plan a career track that would set me up for a PM style role investing in public markets. I'm 25 with somewhat untraditional credentials - I have a CS degree from a top 25 program (globally) and work as a financial data engineer at a large US pension doing front-office portfolio analytics. Was hoping to get something in the private sector but the job market has been extremely tough for recent grads. 

Given my goal, I'm unsure what type of experience/qualifications I should gather over the next 5-10 years but here are my thoughts so far:

In terms of academics, I'm open to getting an MBA or something technical e.g., machine learning or data science. However, I'm unsure how beneficial these would be given I'm not interested in technical research. I also don't want to silo myself and get passed on for management promotions in favor of people with MBAs or more traditional finance backgrounds. This led me to think that an MBA could be more closely aligned with my vision, and could compliment my CS degree nicely, but it is unclear how necessary it is. I'll add that I'm most passionate about investing and doing investment research, which is also more aligned with an MBA, but again, unsure if it's necessary. The financial cost and opportunity costs in terms of salary and experience are the main cons here.  

In terms of work experience, there's a clear route into HFT which pays well, but potentially strays from my goals. Another consideration is quant analyst or a technical equity research role at a HF or investment bank, but I'm not sure if I could crack these yet as I'm unsure how my public sector experience will be viewed.

As you can likely tell by now, I have way more questions than answers. Any and all advice is appreciated. Thanks for bearing with me.