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After few good years, EMI (do not want to publicly display the name of the bank) suddenly ask for a document that does not exist

Amo, you are heavily misinformed and, I must say - in trouble. Two most important notes:

You do need EIN as you need to fill at least form 5472 (About Form 5472, Information Return of a 25% Foreign-Owned U.S. Corporation or a Foreign Corporation Engaged in a U.S. Trade or Business | Internal Revenue Service) every year, and paired form 1120 (About Form 1120, U.S. Corporation Income Tax Return | Internal Revenue Service). Otherwise you are looking into 25k fine per year.

Act quickly, talk with tax advisor...
 
i think to use the us is extreamly dangerous, any formal mistake , or any complain against your company fall under us jurisdictions that make you very vulnerable. better close it and find another place, there is plenty