We've used FAM to conceive all 3 times, so I knew when I ovulated for #1 and #3 exactly, and walked into the MWs saying, "I'm due on...." they have always said, "Mmmm, okay." With #2, DH went out of town for work for 3 weeks, so I figured there's no way we'll get pg this month and stopped charting. Except, oops, we got pg that month. so then I had a 6 day window where I might have ovulated. My MW got out the "date wheel" calculated from lmp, and added 6 days! She was like, "how does the 15th sound?" I said, sure and guess when he was born. the 15th!
Anyway, all that is to say that I think I'd go w/ the later date b/c I'd rather have the "insurance" on the other end, but I have absolutely no familiarity w/ IUI procedures, so I might be speaking out of turn...although PPs have said they'd go w/ later, too.
At this point, i would totally share your frustration, too, though. I absolutely do not want to have this baby in July, despite my due date being the 30th of June! I'm counting on never having gone past my due date to get me through!