First of all, I'm very sorry for your loss.
I asked a very similar question as My Father passed away - next test(s) on what remains of his DNA sample?My Father passed away - next test(s) on what remains of his DNA sample? a few months ago and the answer was extremely helpful. I'm happy to say that I was able to perform everything on my Father -- mtFull, Family Finder, Y-111 and Big Y all successfully. Everything but mtFull was done after he was deceased. However, a big part of the reason for that was because my Father had four vials of samples as opposed to just two. This is because he tested through the Genographic Project originally and at that time they handled his maternal and paternal lines separately, which thankfully yielded four samples.
Anyway, that doesn't help you much if you only have two vials of sample to work with. Do call FTDNA and find out how much sample is left for your cousin and see what they suggest. Not always, but in general I've found their staff to be very helpful.
There are other people here who know far more than I do, but if after doing the Big-Y you take it over to YFull, they can usually give you a pretty good Y-111 using their granular system, without using up any more sample. If a Family Finder from your cousin would be very relevant to you, you may want to do that and a Big-Y, knowing that YFull can help in terms of a Y-111. Of course if there is any sample left over after doing those two, you may be able to do a Y-111 anyway at FTDNA.
You really need to think about what would be the most helpful to you and important to get. If there is truly no other living person who can provide a Y sample, that would be pretty important in my book. At the same time, Family Finder from the right person can be invaluable. Definitely choose which test you're going to do very carefully.
Another note: whatever you order, do it by phone with FTDNA and make sure that they know that the subject has deceased. I did that with my Father and so every time I spoke to them at the different stages of testing, they were well aware that he was deceased and I'd like to think that they took extra care at the lab, etc. because they knew that fact.
Anyway, best of luck. It's a big decision and I certainly can sympathize.