Tom,
I am proud of you to consider doing travels like this. Honestly, and I don't mean this (or anything else I write) in the least bit negative way, I know very little about your conditions so what I say may not apply at all.
First, I would seek other people in your condition who have done similar things, i.e. has anyone similar doe John o'Groats to Lands End? Talk with them. What about hiking the Appalachian Trail or its equivalent in your area? Perhaps ask this on a forum for people with your conditions. I would think those that have the conditions would give a much clearer answer than those without. Sort of like asking those who actually lost 100 pounds how they did it instead of asking a guy who is trying to lose 100 pounds.
Here are my thoughts. Riding solo on the TA is no big deal. Lots of people have done it for decades. I personally have crossed the USA 6 times solo. You must know how to keep your self from getting lonely. I talk to the locals, read books, write letters (who does that anymore!), send postcards (if I can find them), etc. Over the years it is a bit harder though as after 45+ years of touring, the excitement is not the same as in the early days. Again, solo is quite doable.
As far as the insomnia goes, can you be productive during that time you are awake because you can't sleep? By this I mean, are you just super tired and can't do anything really. Or would you be able to say start riding again at 4am and then if tired at 11am, take a nap in a park? If the latter, perhaps you could ride (with breaks as needed) and just setup, cook, etc. as your schedule permits. I would guess so but again, I am clueless of what the details of insomnia entails other than hard to sleep/stay asleep.
Based that you are into extreme sports, are well above average fitness, and life-long cyclist, I totally think you are able bodied.
I don't think some of the usual negatives like being in a foreign english-speaking country (other than riding on the wrong side of the road
), crazy times (you are not a US citizen so can't relate), or the pandemic (who hasn't had to deal with it so nothing new) will stress you that much since you are into extreme things, i.e. the usual stuff may not bother you. I get a sense you are just concerned if YOU can do it. I would suggest a couple of things.
1) Don't do a fund raiser. This trip is about you. There definitely might be a stress about dealing with the fundraising aspects that is not needed. If you want to do a fundraiser, change the criteria to miles ridden, i.e. pay "x' pence per mile ridden, number states ridden in, etc. The goal is in the doing, not the finish.
2) Take as many or as few days off for "rebuilding" as you need while on tour, going solo greatly allows you to do that.
3) Change the focus of the trip from I want to cross the country to I want to start on this route and see how far I can get, regardless of the distance. If I get across the country, great. If I to a point where the enjoyment stops, I can rent a car, hop a bus or train (not like they are in Europe!), and see other parts of America. This removes the stress of having to "complete" the trip and just have a journey or a wander.
4) Try to not have a fixed return date or if you must, definitely consider #3. You may only get to Kansas but then you could always come back another time and continue from Kansas.
You say you have doubts but you don't really say what those doubts are. Most people doing any 3+ month adventure have doubts if they are honest. The doubts are just different for everybody.
I totally think you are physically capable of doing it. The 3 months will pass fairly quickly. After all probably 100k people have done this over the decades. I just don't how bi-polar with insomnia factors in because I don't know how this affects you and the ride.
If you decide not to do it, that is not a sign of a failure but perhaps one of wisdom since you chose not to go knowing conditions might make things overly difficult for you. For instance, I am immuno-suppressed due to a liver transplant. Therefore, while I would love to tour in certain parts of the world. While I COULD tour in say India or the 'Stans, the risk of getting extremely sick or dying makes the those locations not worthwhile to me. Doesn't mean I am a failure but just something I probably shouldn't do.
Tailwinds, John