I submitted a bunch PhD applications for US programs, but a very important letter of recommendation didn't make it before the deadline. I have the required number of letters, but for this one letter I received the confirmation from the professor just a day too late, so I couldn't include it in the application. I know for a fact that this letter could make all the difference, because it answers the most important question about my background the committees might have, while the other letters can't. This letter is also my most up-to-date letter in terms of how recently I have worked with the recommender.
I know I should have tried harder to make sure this won't happen, but it happened now, and I missed to include this only one day too late. Actually the recommender had already written the letter for me and submitted it for a few other applications, but I had only talked to the professor about those applications and did not have the confirmation for additional applications yet, so I thought it's rude to send invitation from portals before getting the confirmation (in retrospect, maybe I was wrong).
I contacted the programs about this right away to ask if they can allow me to add this letter, but unfortunately the response was negative. But I know for a fact that this letter could really turn a 'No' into a 'Yes', and I want to do everything I can to get this letter to the committees without going too far that it'd instead hurt my chances. What can I do? Is it appropriate to contact professors I mentioned in my statement directly to ask them if they would accept the letter via email from the letter writer? Or try to contact someone else, other than the listed "admissions contact", who already refused to include the letter, and ask them if they can help with this?
Any suggestions would be very much appreciated. I just want this letter to be considered in my application, without crossing a line that'd make things even worse than not having the letter.
(The country is the US and the programs are engineering/computer science programs)