Wow! Flavo, you have one of those wonderful problems!
Personally I would not cut it yet unless you need the wood. As you stated, it can be nice shade for the horses and they do need shade. It is alive and that is why I would wait unless the wood is needed. For sure felling that thing will be about as simple as can be.
So, how do you handle the wood once the tree is down? That you will find, is very simple and easy....if you do it right. Look at the following picture.
Notice I've unloaded the wood and all that is left in the trailer are the tools I take to the woods. In the rear of the trailer is one tool I really hate to be without. See the wood handle and the blue on the end of it? That is a 4' cant hook. Using one of these really makes moving big logs really easy. Now look at the next picture. Do you see the cant hook? How do you suppose I got those logs onto the dray? Very simple and easy with the cant hook. I learned many, many moons ago that a small man can move a huge log using one of these. When I was logging, I weighed around 155 or 160 (oh to be that again!) but I could move a 3' or larger diameter log that was 12' or longer all by myself without straining. Child's play.
So let's say you load the logs and haul them to your work area. How do you split them? First, please understand that just because these things are big does not mean they will split hard. I like sledge and wedge but a splitting maul will work if you use it right. Just don't hit in the same spot all the time. Strike the log on the far side and then strike it in the center and then towards your side. Sure, you have to strike it more than once but who cares? In the end you will not strike that thing any more than you would smaller logs to get the same amount of wood. I actually think you will split faster doing these big logs and I've never minded doing them.
Naturally, after an injury and with age I've graduated to the hydraulics and still find no problem splitting these things. Of course one does want to be sure to do it right which begins by setting the splitter in vertical mode. Sometimes a prybar or the cant hook can be very handy for turning them once on the splitter but again, if you do it right it is not all that difficult. Even with my back back I can split some mighty big stuff without straining myself too much.
Funny that a neighbor thought these hydraulic splitters were junk, but one day he finally asked if I thought our 20 ton splitter would split some huge white oak rounds. Of course it would! I took it to his place and although we worked together to wrestle them around, he was so amazed at the ease of splitting that he purchased a splitter the very next day.
Don't fear the big stuff. You will probably come to love it once you get the hang of it. And that is some fine looking wood you have there.