I don't have experience with Hog Peanut, so I won't comment on that one. Trumpet Honeysuckle is a great vine for your region - probably the least aggressive spreader of the bunch, and also probably nearly evergreen, which is nice in the winter. It is a twining vine that needs something of small diameter - like a wire fence, bushes, small pole, string, etc. - to climb. It cannot climb a shed wall. Virgina Bower does tend to be aggressive, I think. I have never grown it, but see it wild frequently. In my opinion it isn't as nice a vine for the garden as Trumpet Honeysuckle. I have a ton of Virginia Creeper here, and have never heard about the berries being poisonous. I wouldn't be concerned about the berries, which are a favorite food of many birds. Virginia Creeper is a little aggressive, as are most vines. Another choice I like is trumpet creeper - Campsis radicans. This is an agressive spreader, but has very showy flowers most of the summer and fall and attracts hummingbirds, making it a really nice addition to a yard that has roon for it. If the shed is surrounded by lawn or other open space where it will be easy to contain the root suckers of Trumpet Creeper, then it sould be my first choice. Another option is Carolina jessamine,which has yellow flowers in early spring. Also Cross Vine, which is another flowering vine. Jessamine is a vigorous, evergreen vine. Cross vine tends to be a little more sparse in growth habit, but is an interesting addition to the yard.

I grow several grapevines up against chain link (hurricane) fencing. It is a great trellis. BUT you need to keep the vine's trunk on one side of the fence and I enforce this with winter annual pruning. If you let the vine grow through and stay it will expand eventually stretching (damaging) the fence and girdling (damaging) the vine from that point upward. Unless someone did this you can almost assuredly expect the vine grown though the fence possibly at several places. Things you can do: (optional) If there is part of the vine running along the ground (particularly if the growth is not yet woody) you can cut and remove a half section of bark and bury it in a pot. It will root. Kind of like doing an air layering. Eventually you cut the vine from the mother plant. Then you have a potted grape vine that can be transplanted to the new desired area. (optional) You can take dormant cuttings, callous them and then plant them either directly in the ground or pots, again to produce a new vine. Either preserves the vine in case something goes wrong. As to the old vine and the fence: The vine is likely fairly woody so it will stand on its own once the fence is removed. But it is not used to being unsupported so it will be somewhat weak. If you do not mind cutting up the fence in that area where it is attached (it can be weaved back together with minimal loss of length later if desired) and assuming the worst kind of entanglement between vine and fence then it should be a breeze to do what you want. I'd wait until the leaves are gone so you can better see the issues. And being South Dakota, I'd wait until winter is pretty much over but the vine has not broken dormancy to remove the fence. Wire cutters for the fence; I like the wire cutting notch in pliers for this job. Cut parts of fence can be twirled up unwinding the cut wire. May want to do SOME pruning first to get rid of some of the vine that is in the way. And to cut as many of the tendrils wrapped around the fencing s you can reach. BUT do keep in mind there will be parts of the vine you want to keep and the manipulations you do may break some of the desirable parts necessitating secondary backup parts. So leave more vine on than you might otherwise in case some gets damaged in removing the fence. Pruning and metal cutting will be back and forth until the fencing is finally clear. If you have an old pair of pruning shears you may want to use them as cutting tendrils results in a few times where you are actually "cutting" the fence with is really bad for the shears' edge. Once the fence is out of the way evaluate as to how much support is needed. Also look for girdled parts of the vine. Either do more pruning or bring the new trellis into position. Attach trellis to ground. Use nursery tape to tie the vine to the trellis. And start pruning for that years production and next year's wood. Once the vine breaks dormancy the tendrils will secure the vine and eventually the nursery tape can be removed. https://www.sdstate.edu/ps/research/viticulture/grapeproduction.cfm

you may have used the gw google, but in the event you have not, here is a start to some interesting/informative reading. Also, read al's 5-1-1 [water movement] threads like this one in this forum. grapes, grapes, and more grapes ...

There is a fruit and orchard forum that would be more likely to have answers. https://www.gardenweb.com/discussions/fruit