I got told by the imaging place you could get it done between 11 and 13 weeks.
They actually gave me a date I could have the scan which was like 11wks 3 days but I went when I was 12 + 4 so DH could come too.
If its only one day, I dont get what their problem is. I'd be ringing and telling them that is the ONLY day you can go otherwise you can go at all, and you really want this scan etc. Or try a different ultrasound place?