If tstop4me's advice doesn't work for you:
You presumably fall towards the inside because the structures inside your body don't apply equal pressure and/or tension to both sides of your feet. That's just basic physics. (Except: much less experienced skaters than you are sometimes afraid to go onto an outside edge, because they are afraid of falling. It's easier to stop a fall towards the inside where they can put down the other foot to stop what would otherwise be a fall, than it is to the outside, where you need to use more complicated ways to restore balance. But I assume that's not true for someone as experienced as you.) Various medical sources say it might have something to do with the length or structure or physical properties of the muscles and ligaments and other connective tissue inside your body.
Those internal structures must apply more pressure to the outside edge of your feet than the inside edge- at least when you put your full weight on a foot. I.E., if you don't put much weight on a foot maybe you do get roughly equal pressure to both sides of your feet, but under full body weight, they might collapse unequally.
I believe I don't have flat feet (yet - a lot of people develop it eventually, as their muscles and ligaments and other connective tissue gets older). But if I did, I might try to overcompensate a bit, and setup the insoles of my boots in such a way that tends to push me onto the outside edge just a little bit. E.g., put some strips of tape underneath the insole on the side of the boot you tend to fall towards - in this case the inside.
Essentially you would be compensating for an unequal pressures and/or tensions inside your body with an unequal pressure outside your body.
I don't have the exact same problem. My problem is that the bottoms of my feet are tilted slightly right to left, I think because one leg is maybe 1/2 inch shorter than the other. It's not enough to notice, walking, though I suppose it is possible a very skilled podiatrist or performance dance teacher would have been able to see it. (The podiatrist I went to, the one my insurance company was willing to pay for, couldn't see it. But she wasn't one of the two podiatrists in a multistate area around me that a lot of skaters and dancers love.) But balancing on an ice skating blade, it becomes quite noticeable. In addition someone misfit me for skate boots, in part because they had me stand on impressible foam while on both feet, instead of balancing on one foot. They are full custom boots, but the boot maker based their efforts on those foam impressions. It was literally the case that gliding on one foot, in my boots, the right side of my feet weren't in contact at all with the bottom of the boot, rather than the equal pressure they should have had. It took me a lot of time, as well as conversations with other skaters and skate techs, and reading material that podiatrists have written, to even realize that was the source of the problem. I had to become consciously aware of the sensations of pressure on my skin at different parts of my feet. Something I hadn't thought to do before.
I like cloth athletic tape, sometimes called cloth first aid tape - a good quality brand is Johnson & Johnson, and it really does make a difference. Some of the other brands, and almost certainly dollar store tape, and even more so duct tape, have unstable adhesive, which tends to gradually work its way out from under the tape. So other brands might work for a few days, but not much longer. Eventually I got into instead cutting my self a custom insole out of foam, which is a bit more stable, though foams eventually collapse. I use cheap foam, from a camping mattress pad, which has the advantage of being skin safe. But in theory other foams, like a good urethane foam, would last longer. If I recall tstop4me once mentioned trying other foams, and covering them with a skin safe layer.
I got the foam idea from a skater I corresponded with who had extreme problems with pronation. She was cutting carpet foam. But she had to do something else, which sounds a bit extreme. When her feet collapsed on the inside edge, that meant her feet would spread out on the inside edge. (Actually, pretty much everyone's feet expand outwards a little when the collapse to absorb weight. But hers tended to expand much more on the inside than the outside.) So on the side of the boot next to that inside edge she added adhesive padding (something like "moleskin" - though once again maybe one should avoid the dollar store cheap stuff, though it might be good enough for a short term experiment) so the feet could not collapse, because there was no space to expand into. As far as I've tried to find out, that is a very unusual thing to do, which most podiatrists don't usually do. So maybe it is NOT the best possible idea. But, combined with the cut foam insoles, it worked for her.
A real expert might be able to find a foam with the exact right properties to compensate for the unequal collapse that occurs under different amounts of weight, and make it so you would create equal pressure under both sides of your feet, at all stages of putting weight under your feet. I don't know how you would do that, other than experimenting with foams with different degrees of stiffness. If you deal with doctors much, you may eventually find they often to take an empirical approach too: If one solution doesn't work, they try another.
Other people have used cork insoles (like some alpine ski stores sell or install in boots for similar problems). They heat the cork, put their feet in the boot, perhaps under full weight, and it molds into the shape that creates close to equal pressure everywhere on the bottom of the feet. I haven't tried that.
I tried a heat moldable insole made by SuperFeet specifically for skate boots, probably roughly along the lines of the heat moldable orthotic that tstop4me mentions. But it was too stiff for my boots. It eventually pushed the sides of my boots outwards, and made them too big near the footbed, so I needed to snug that up and stopped using it. And it rocked side to side inside the boot, as I changed edges, especially once it had stretched out the boots, so I had to compensate for that. Eventually I gave up on stiff molded insoles, and moved to foam. But my boots were "soft dance" skate boots, and maybe that wouldn't apply to good quality stiff freestyle boots? Or at least not before the boots broke down in other ways?
But some podiatrists use similar orthotics in normal shoes. (AFAICT, an orthotic] is just an insole prescribed or created by a certified doctor or orthotist. If you can't find a podiatrist that the skating community loves. Maye It's worth posting in forums like this to try to find one to try to find one close enough for you to drive to in a day or so. That sounds crazy, and traveling that far and staying overnight a day or two at a hotel is expensive. But if you think of all the money you have spent on coaching and ice time trying to compensate for your issue (work it out - it might well be thousands or tens of thousands of dollars), and the personal frustration and abandonment of dreams it has perhaps caused, maybe it's worth it?)
Eventually figuring out how to efficiently compensate for my issues became a fun activity in itself. If you have an analytic bent, you could see it the same way, and be pleased with your ability to do so.
