Doesn't mean you should use the wrong term.
You're trying to use "transrational" to mean "outside of rationality". But "trans-" as a root means "across" or "changed". The only context in which it gains the meaning of "outside of" is when your other root implies a boundary or movement: words like "transcendent" = "to climb across".
"Transrational" does neither. You can't simply borrow a prefix from one context and apply it to another, hoping to gain the implications of the original word.
When you get a blood transfusion, thank your lucky stars it does not mean "to pour out". Ditto for transplant. Perhaps that was the "complication" in Poincare's surgery...
