If it is welded wire I don’t think the vertical wire is very strongly attached to the horizontal wire and if you hook onto the vertical wire it will probably just pull off, but it may be stronger than I think.
If I was trying to bundle the end like I suggested I might just cut through the vertical wires on the first few and then twist a couple of the horizontal wires at the dirt level together and see if that works to get it out of the ground. How deep is it buried now? When going into service here they are usually about 6 inches deep but they get much deeper when they do the job they are supposed to do.
Another idea is to wrap the wire around a piece of metal or wood and then try to pull that instead of the wire.
i don’t see how using anything to pull it vertically would work but I guess you could try a farm jack to see if that works and then use the tractor FEL to pull it vertical by jamming the bucket teeth into the ground and then curling the bucket up to catch the wire. You might have to do a lot of short sections to get it out? Sounds tedious and a lot of back and forth but I think you have a HST that might help with that.
good luck with it, and I think you are too late to avoid the heat but not the real heat of the summer (which is what you you are really looking at avoiding).
I have trouble with 90 degrees when it comes the week after it was 30 for the low…but I will adjust over the spring, hopefully it will be a somewhat gradual increase now