Well, Konstantin died of cholera at Vitebsk in OTL, so it stands to reason he could manage at least until the 1840s (maybe). As to the heir, I have no idea? Maybe naturalizing one of his illegitimate sons (Paul Konstantinovich or Konstantin Konstantinovich) might be the way to go. It would get bonus irony points if either marries Anna Feodorovna's bastard daughter

Speaking of, does anyone have info on Konstantin's three bastards? I know about Pavel Konstantinovich Alexandrov, but the other two listed by wiki, Konstasia Konstantinovna and Konstantin Konstantinovich? Any info at all would help.
Also, would the Greeks accept a legitimated bastard ruler?
Britain might ease off about Russia and Greece coming into personal union if it's clear that that isn't possible, so maybe as a wife for Konstantin's legitimated heir, someone British, or at least part-British. Or Konstantin nominates one of Nikolai's younger sons as his heir.
Joanna might have to convert though. I can't see them allowing her to be openly Catholic in an Orthodox country - didn't go over so well for Othon.