Say I want to travel to Russia on tourist visa, physically visit some Russian bank and open a new account in roubles. Let's say my name is John Doe. Do I write "John Doe" or "Джон Доу" in bank agreement papers?
Any kind of legal documents at a minimum are going to be checked against your passport and whatever other documents they may require in order to authenticate your identity. So, fill it out with exactly the name that your passport has.
There might be a need for other documents- can’t really comment directly on what they may be, but in general there might be a need for a certified translation or additional certifications.