Hmmmm, not wanting him dead rather rules out nuking him from orbit.
The opossum site that Daffyd posted has a page on discouraging opossums from visiting your yard.
Is there something that you could put on top of the fence that would make it difficult or unpleasant for the opossum to walk there?
Like lava?
Not just scavengers, but also beneficial predators. As your link points out in part they will also eat your slugs, snails, mice, moles, voles and rats. They don’t harbor any significant zoonoses.
About their only objectionable activity is a tendency to get into gardens and eat your tomatoes and thump-thumping in your attic if they decide to den up there ( just don’t give 'em access ). That and they apparently skeeve some people out :p.
But as urban wildlife goes, they’re better neighbors than many.
Mama Plant used to keep the neighborhood cats off her cat by putting pepper on the hood and trunk. She bought the cheapest brand in large boxes. 
They got it on their feet, and licked it off.
I suppose lava would work. I was thinking more in the direction of some chicken wire. Or something that smells objectionable to opossums. Or some of those cheap tin foil pinwheels- the motion and flashing reflections of those supposedly scare deer and rabbits, maybe it would scare opossums, too.
Cats is weird.
I use chili powder. It’s much finer than ground pepper.
And by the way, did she sprinkle the pepper directly on the cat? Or just where it hung out?
I use a Co2 pellet pistol for those extra horney bastards. It doesn’t break the skin but lets them know what the price is for their efforts.
I’ve had my share of possum problems. They’re transient, so unless there’s something attractive about your yard, they should move on. My Animal Control considers them vermin - ergo - not their job anymore than mice. Catching it/them and driving it out to the country is probably the best bet.
Call around to various animal orgs, one of them might loan you a trap free of charge, but renting one for $5 doesn’t seem that unreasonable. Check the fees your local vermin removal companies will charge you.
There are lots of possums around here. The Rat Terrier cleaned out one whole family of them a few years ago before I could even find out where they were living, and there was a HUGE trophy possum road killed in the street in front of my house just last Saturday night.
I have gotten really soft about killing anything in the last few years. There was a time that I would have killed any possum on sight just because I didn’t like their looks. I did set a live trap duing the batch the Rat Terrier killed, and caught a cat. Even though I still see possums in the neighborhood now and then, I haven’t had to do battle with them in a while, but I bet the dog has some interactions with them that I don’t know about.
As for the OP, maybe hit that thing with a water hose every time you see it. I bet that it won’t take many good soakings for the possum to figure out that your fence = wetness.
Opposums eat virtually anything, this is why they hang around dogs if you leave food for them. Opposums love dog food, especially dry dog food. They are clever when they find a place they like, such as an attic or garage. They are about equal if not a little worse than raccoons.
Opposums like Racoons know all about pet dogs. They are smart enough to know what they can get away with. They may run from a dog, but if there is dog food or other food or garbage they can get into, they will simply bide their time till they learn the dog’s schedule and adapt to it.
The opposum knows if it’s on a fence the dog can’t get at it, and will sit there biding it’s time. A big dog can’t be hurt by an opposum, unless it’s rabid or it may give it fleas.
My neighbor had an opposum in her garage, what she did was turn the car on and close the door. After about an hour she went in and found a nearly dead opposum which she took care of with a hoe.
They get on Mama Plant’s car.
The hell is this stuff on my foot? They lick it and quickly learn it comes from the car and leave the car alone if they smell pepper.
Re: Shiny Pinwheels. I dunno, having more teeth than brains, they may be hyp-no-tized and just sit there staring.
Note: if you trap and move a 'possum yourself, make sure to use plenty of some kind of disposable barrier, e.g. newspaper, between the trap and anything you wish to continue to own, such as the interior of your car. Possum poop is shockingly stinky and sticky. That shit is supa-skank.
BTW, despite what some might think after looking at the site **Daffyd **linked, there is no immanent paucity of 'possums. They, like coyotes, raccoons, and brown rats, have adapted all too well to urban environments, and are a “species of Least Concern” endangerment-wise. While they are indeed valuable scavenger-predators and part of the Circle of Life, and while they are as deserving of humane treatment as any creature, we are in no danger of running out of the little bastids.
No suggestions, but I thought your thread title would make a great country song.
Maybe y’all just live in a place that has nice, friendly possum-wossums, but around here, ours are mean, aggressive and nasty. Wouldn’t be surprised to see one picking its teeth with razor wire…except I don’t think they’re anyways near that clean.
I once lost a beloved pet cat to a possum that just eviscerated him in one swipe. They grow quite large and fearless–of dogs and people both. My husband once faced off with a big one in the back yard who was not the least bit fazed by either my husband or the big stick he was shaking at him. So for all those telling you that possums aren’t aggressive and should be welcomed into the fold as handy helpers who keep your backyard free from small rodents…well, I’d rather have the rodents.
I live right below you. I’ve never seen them vicious, not that I doubt you. Maybe there’s something in the water down here.
Really? Honestly?
One of your neighbors has a gun and will be more than happy to prove his manliness by disposing of a possum for you.
A couple of years ago I had a girlfriend I could not talk out of leaving out catfood for strays. Eventually, we got a Raccoon, which I named “Coony.” I’d be practicing my guitar or smoking the hookah with my window open and sure enough, right at 10:00 every night I’d hear munch munch munch
I’d slowly peeeer out the window, look down, and there was his fat little self chowing down on catfood. So I’d go get this big 6’ shower rod I kept near the door, throw it open and yell HEEEYAAA, and chase the little thing, half naked and barefoot, for a quarter mile, poking him with a huge metal rod the entire way, until he found a tree to run up. We kept that routine up for about a month before he finally went away.
Maybe we just grow them hard-bitten and cantakerous up here, whereas down South, you all have much nicer-mannered vermin. Maybe you have Deep-South-gracious possums who invite you to share in a mint julep out on the verandah, and ours are mean-as-a-snake hillbilly possums with a three day drunk goin’ on and an itchy finger on the shotgun.
I dunno.
So what you’re saying is that up North, even the critters are rude. Good to know. 
My first instinct on reading the title was “Eh, it’s a possum. The dogs’ll run it off. Or hell, she’s got setters. They might kill it instead.” Evidently, however, her dogs are bigger, squishier wimps than mine are–our 55# shepherd killed one in the yard one night and neither of us had any idea until we found the corpse on the patio the next day. While I wouldn’t send an 8# cat against a possum, there’s really not a terrible lot of damage one can really do against dogs the size of the OP’s. Scrapes on the feet and legs are by far the most common injury incurred in scraps with possums.
Possums are vicious bastards when cornered, but by and large they’re not going to launch themselves at you or anything. (Of course, all bets are off with the behavior of rabid wildlife, but possums are not huge rabies vectors.) So I wouldn’t worry about letting the dogs out, really. Especially during the day; possums are nocturnal.
Maybe I’m stupid, but I don’t see the problem with this. No, scratch that; I see why you’d be vexed at being separated from your dog, but I think it’s worthwhile. Opossums CAN carry rabies, and rabies can be transmitted to other dogs and to humans. A quarantine of a few weeks to determine whether your dogs are infected is quite reasonable. Rabies is rare not merely because of the vaccine but because of aggressive measures such as those you decry.
I guess California possums make for happy possums. Maybe they get into the neighbors ganga, but I just grab 'em by the tail and toss 'em out of the yard. They come back saying “dude, what was that all about?”