There is a moratorium on players being traded after they have been recently signed (unless its a case of sign and trade scenario), but this is merely a time formality and not a real obstacle to the transaction. If signed in July, Love could be traded before the end of October?
In terms of how their games fit together, I do believe Love is a better fit for James than Cousins. However, your argument on personality fit, to my eyes, is slightly weaker. Love is very much a diva in his own right, and Lebron (while I agree is imperfect in many areas of character) is less of a diva than you describe. I do not believe the chemistry issues with Cleveland would be demonstrably worse, or better, with Cousins than they were with Love.
My opinion (as amateurish as it is) is that Cousins would defer to Lebron (and perhaps even Irving), and provide a defensive, rebounding, and offensive threat superior to the Mozgov-Thompson combination, and provide the ability to carry the offense sans James and Irving, saving their legs throughout the season and during deep playoff runs. Not that Love could not have accomplished the same to some degree had he been present and healthy. A Cousins - Irving pairing is also a combination that can carry Cleveland deep into the future as Lebron ages and continues to play past his prime.
A combination of Love + his pick of Mozgov, Thompson, Shumpert, Haywood's expiring, future picks, would be far easier to sell to Sac fans and ownership than a half-dozen unknown draft picks in mid range, an expiring contract, and two or three promising 3rd-4th year players.