I've cut down many hardwoods in my grove like this, just keep cutting, know where the tensions are, be careful, don't have anyone or any machinery around.
You don't have the experience or reflexes for that. Cool.
Then I'd wrap a chain/ cable around the tree as high up as my longest ladder can reach, and pull it down. I'd want the chain or cable longer than the tree or any tree it is leaning on - my great uncle lost his leg when they pulled a weak little limb off a tree - tree was hollow & whole tree fell over on him, pushed him under the tractor wheel. Chain was too short. Needs to be longer than the tallest tree involved in this, not just the branch or tree you are working on. Family learned a hard lesson.
If it looks like it is hung up too much for this to ever work; your tractor isn't big enough to pull it; it looks like the propane tank or something else is in peril; I would hire a tree person with a bucket to come out & get it on the ground for me. They have the insurance & experience to get it done for $600 or so, no worries.
I would not, under any circumstances, try to pin down the tree with a backhoe or by chaining it to other trees. _THAT_ will get you killed. There are many stresses & forces involved, in different directions, that can be very very powerful for a couple feet. You do not want to try to prevent these - they will blow something apart. You want to plan for them and allow them in controlled directions. The goal is to get the tree laying flat on the ground - not swinging like the I beam in the pickup commercials back & forth from another tree. Your backhoe can be damaged. You need to accept that the other tree could fall over & plan for that - unlikely, but you need to plan for it. Chaining it to another tree - you could end up with the tree hanging up in the air by the chain - now what??? Or the other tree gets pulled over, now you have the leaning tree falling away from you, and the 'good' tree falling towards you, and the other tree that is being leaned on also falling over or shedding branches...... Any tree involved in this can fall over, unlikely as that is, or lose a branch to fall down upon you. You don't want to involve more trees - you want this less complicxated, not more complicated.
I don't like involving anything else at all; or a long chain/cable and going one direction with it.
Trapping or preventing a spring loaded tree from moving is not a good plan - it _will_ move. With experience you can plan for the directions of movement. Trying to pin it down, you aren't sure what it will do with it's pent up energy.
My brother in law has a similar hardwood tree the tap root brok, leaning in the general direction of the LP tank as well. It is leaning on a very old oak they want to save. My BiL has all sorts of ideas of how to take it down.
YIKES he scares me!!!!

So it has sat there for 4 years now, time bomb...
Me, I'd bring the big tractor, long cable, pull to the side, cut some cuts in the trunk, and pull it to the side. If a limb gets damaged on the big oak so be it, most likely it will not.
My way, everyone, including the LP tank, is safe. Only bad thing is the big oak could be slightly damaged.
BiL's way, oh my gosh, I don't want to be in the same county trying to use a forklift or small crane he owns to try to control the tree by being near it with those things.... Butt of the log will roll around onto the person with the chainsaw; and the tree will fall onto the forklift or pull the crane over,; kills both of us! Would never want to be involved with something like that. I'd rather just do it free-hand then, see if it misses the LP tank & have room to move, and sort of know where things will fall. No way do I want machinery or chains holding the tree 'in place' - the tree is going to go someplce, and it's neighbor might as well.
Glad you know your limits. If it takes this much anylizing to figure this out - hire someone to do it for you. Whatever you do - I would not be at all comfortable trying to trap the tree into not moving.......
--->Paul