BTC-e international wire transfer in USD from Russia
I contacted another bank and was reassured that any wire transfer from a Russian resident to any person or company could be made without supporting documents if it is under 5000 USD or equivalent.
As for the reason of the payment it could be filled with something like "transfer of personal funds"

Edit: However, the so called currency control, which is an inner bank department, demands documents anyway. Normally it's a customer agreement which could be simply downloaded from the website and some kind of a document showing that you're an account holder.