I think the rejection from LNF is valid because they need time to process. From my experience it takes them from 4 to 20 hours to process a claim. If you are in such a hurry you should have called them and tell them to look at your case sooner, but again you didn't give them enough time to process your claim.

Next time, do it a bit earlier, and if they reject you once, you have 1 chance to appeal. If they reject you again, my advice is to wait it out and try again in about a week's time. LNF dept will stop looking at your case after 2-3 appeals and they are not the most customer oriented people in Marriott. It's not nice but I do think that LNF is a very generous offer from Marriott.