Scholarship asks for copy of diploma, old university slow to print it
Calm down

Getting angry at your university and/or its staff is tempting but solves nothing; if anything, it will make them less inclined to help you.  Focus on solving the problem and being respectful to everybody, even if you don't want to.

In particular, it will help to act as if everybody involved is doing their best in good faith to help you, but may be unable or unsure how to do so.  Don't treat them as though they're just trying to obstruct you, even if it seems like they are.  Don't focus on the amount of tuition money you've spent; that's not going to seem like a relevant issue to anyone who can help, and it might be a distraction to you.

Understand the problem

At the root of your situation is the fact that you're stuck in the middle of a cultural misunderstanding.

In US academia, the diploma is strictly a ceremonial piece of paper. Nobody uses it for anything except to hang on the wall.  If you want an official document that verifies your degree, you get a transcript instead.

As such, within this academic culture, there is no reason why anybody should be in a hurry to get a diploma, so university systems won't be set up to provide it on any particular timeframe; they'll schedule production of the diplomas in a way that is most convenient and cost-effective for the institution.  This most likely means producing them as a single batch, perhaps by an outside contractor.  Trying to do one as a one-off may well be quite expensive and/or a major disruption to their existing process.  It's not necessarily a simple procedure that they're just putting off because they're lazy or uncaring.

However, my understanding (based in part on things I've read in this site) is that in European academia, the diploma is an important official document, which serves as the standard proof of a degree.  Thus, a European institution would expect that anyone who graduates with a degree will get the diploma immediately, and the most likely reason for someone to be unable to show a diploma is that they didn't actually graduate.

I would focus on trying to help people at both universities understand this disconnect, and asking for flexibility from them in helping you navigate it.

Pursue all avenues

It seems like so far, you've been focusing on trying to get your university to speed up production of your diploma.  But it looks like that could be a dead end.  So try something else.

They've offered a transcript and a signed letter.  Take it for sure.  Then, I'd try to get the European university to compromise by accepting these documents as proof of your degree, at least temporarily.  Try to explain to them that these are the standard documents that US universities expect to create and see.

Get faculty involved

It sounds like so far, you've been trying to negotiate directly with administrative offices on both sides.  That's a good place to start, but at the end of the day, they don't have the same investment in your academic success that faculty do.

I'd look for any faculty at your current university who know you, and who might be able to make an impression with a letter to the European institution.  For instance, is there anyone with contacts at the European university, or who at least is familiar with the academic system in that country?  They might be able to help convince them to accept your alternative documents.

Likewise, get in touch with faculty in the European department; either someone in charge of your specific graduate program, or some other faculty member who you've had contact with.  (If it's someone who has some familiarity with US academia, all the better.)  Make sure they understand that this issue, if not resolved, might prevent you from attending their university at all.  They went to the trouble to accept and recruit you; they'll be very disappointed if bureaucratic issues wreck it all.  So they have an incentive to help.

Have a backup plan

Although hopefully you'll be able to get this resolved, there's always a chance that you might not.  Life isn't perfect or fair, and unfortunate things happen sometimes.  But your life will go on.  Make a general plan as to what you will do if this problem can't be resolved and you don't get the scholarship.  Attend a different program?  Take a year off and apply again in a year?  Follow a different path entirely?