Anonymous participant since it's a big online teaching company with thousands of teachers (and surely some of them are here on dtv), I wouldn't do that. I believe your best call is to write a cover letter with what you stated above here, perhaps supported by emails or logs (you can send an email today to your company stating that you need that document, supporting by a screenshot of the mail of the embassy. If they don't give that to you, just take a screenshot and send it to the embassy). Don't think those guys at the embassy are stupid, they see dozens of cases every day. good luck
dude, don't play with fire. Just do the things as they asked you without getting in trouble. There are people who got black listed and got their app denied many times cuz of that. Talk to your boss and explain the embassy's needs BUT don't sign it yourself. that's the easiest way to get in trouble
recommended is not the right term, it is forbidden to leave the country you have applied for your visa at. Leaving the country is a reason to can get your application denied.