Barney's fecal poop looks very much like that of my own buns. My bun Toby, I believe was born with megacolon. He has the spotted rabbit gene, which it tends to be associated with, and I first noticed signs of the strange pooh shape when he was around 6 weeks old. Toby has the oblong poop as well as the very large and very small, and occasionally the very strangely shaped ones too, but no mucous in the pooh as of yet. When I'm feeding him too many pellets, I will start seeing fecal poop that is more moist, and sometimes mushy. For his diet he gets timothy hay that has stalky and leafy bits, as well as a little alfalfa hay for the extra protein to help keep his weight up. He doesn't seem to have problems with leafy lettuce, carrot tops, parsley, cilantro, apple leaves, willow leaves, or raspberry leaves. His pellet amount has to be very specific. I have tried no pellets, but he started having extremely large log shaped moist fecal pooh. I felt like it looked much better when he was on pellets, so I put him back on a small amount. Right now he gets 1/2 tsp, twice a day, but has done ok on up to a full tsp. If I try and give more than that, he starts having moist mushy fecal pooh. Also, different types of pellets have caused more problems for him than others. Pellets with corn products in them, caused the most problems. Right now he is getting a timothy based pellet. On this diet he seems to be doing the best and it is keeping his pooh and digestion stable.
My other bun Zeus, has pooh that sometimes looks a lot like the last pics of Barney's pooh. I don't know that Zeus has megacolon, but does have pooh that looks like a megacolon buns. I don't believe he was born with it, but seemed to develop these pooh problems after several bouts with stasis. I later figured out that it was his pellets that were causing him to continue to get sick. Once I stopped pellets, he got better and didn't have stasis again. But after the stasis I saw that he was having the odd shaped and sized fecal pooh. It was large, small, and some were strangely shaped, but he didn't really have the oblong shaped ones so much. His pooh has been this way for 18 months, so I figure the stasis must have caused some permanent damage to his colon. I did try a few times, to very gradually introduce different types of pellets, but every time his poop would start looking much worse and eventually he would start showing signs of an upset tummy. But when he was off pellets, his pooh would get to looking a lot better and closer to normal looking, and on pellets it would look more like Barney's evening pooh. For Zeus' diet he gets no pellets at all, timothy hay with the stalky bits, as the indigestible fiber seems to help with his pooh the most, some alfalfa hay, and the same leafy greens as Toby. Neither of my buns has needed meds as they have been stable on their carefully controlled diet.
I don't know how it will be for Barney, but for my two the foods that cause them problems are foods with higher sugars and starches in them. So pellets, starchy veggies, grains, fruit, even hay like grain hays, that may have seed heads or higher sugars in them. I've just had to be careful with their diet and try and avoid things that I think could possibly cause a problem, cruciferous veggies being among them. But then it's also a process of elimination. You try eliminating something in the diet to see if it helps improve things. It's a bit of a guessing game trying to figure out if there are any foods contributing to the problem. But you'll learn that you have to do this experimenting very carefully and in very small amounts. Like when I was seeing if Zeus could tolerate pellets again, I was only giving one small pellet to start, and only increasing one more a day. It took me about a month to get up to a tsp. of pellets. And you have to be very observant in changes of body language to catch things early before they can get bad. The earliest indication that I had from Zeus that the pellets were upsetting his stomach, was him laying down and squinting his eyes, which was uncharacteristic for him. It's not always easy sorting out what is going to be the best diet. You can't always go with what works for other people, as each buns problems might be slightly different.
Hopefully Barney's problem is still treatable and this will all clear up with the right treatment.