Hello,

My advice would be to reach out to Bain's recruiters, explain your situation, and ask to be put in contact with their immigration department. Set up a call with them to understand the process and your options. Alternatively you could try reaching out to some immigration attorneys in the US. 

Most immigrants I know who worked in consulting had work authorization tied to their student Visa, so they had studied in the US on that student Visa before transitioning to full-time work. If you are currently studying at a US institution, feel free to message me for more details on this. Unfortunately I am not aware of what the Visa process looks like if you are not going down the student route.

I also doubt that the person referring you will know that you need a Visa unless you explicitly told them - the vast majority of people will not be aware of Visa requirements for anyone but themselves. So it is your responsibility to make sure you are legally able to work at the office you are applying for.

Unfortunately the Visa process has not gotten easier over time, if anything it has gotten harder due to COVID-associated restrictions.

Regarding the choice between applying for a US office vs a European office, I would do some research first to understand whether you would be able to get a Visa to work in a US office right now. If that option is not available, unfortunately there will not be much else you can do but wait until you can apply to offices in Europe. If there is a pathway for you getting the Visa, then I would do more diligence on the respective offices (e.g. what is the work culture like, what kinds of clients do they work with, what is the salary like, etc.) and also think about your personal preferences (would you prefer to live/work in the US or Europe)