See Portugal GV Fund Comparison? for a discussion of using public funds.
Also, on your original question: There is a discussion here Apostille and Document Prep for Indian GV Investors about whether you would need a criminal background check from India. If you do I suspect that would be the slowest thing right now.
Bison Bank seems to be one of the only banks that allows US persons to open an account remotely. You could start the process by opening an account there, or at least schedule a call to get more information. They might be a good point of contact as they work with a number of funds.