Well to start there is a lot wrong with the question you're asking.
You don't just… be psychotic, it's not a conscious decision of "well I guess I'll develop psychopathy today."
Also, psychopathy is not a clearly defined psychological term, or at least it is not an official designation in the Diagnostic and Statistical Manual of Mental Disorders (DSM.) The closest there would be Antisocial Personality Disorder, which is marked by…
A. There is a pervasive pattern of disregard for and violation of the rights of others occurring since age 15, as indicated by three (or more) of the following:
1. Failure to conform to social norms with respect to lawful behaviors as indicated by repeatedly performing acts that are grounds for arrest.
2. Deceitfulness, as indicated by repeated lying, use of aliases, or conning others for personal profit or pleasure.
3. Impulsivity or failure to plan ahead.
4. Irritability and aggressiveness, as indicated by repeated physical fights or assaults.
5. Reckless disregard for safety of self or others.
6. Consistent irresponsibility, as indicated by repeated failure to sustain consistent work behavior or honor financial obligations.
7. Lack of remorse, as indicated by being indifferent or rationalizing having hurt, mistreated, or stolen from another.
B. The individual is at least age 18 years of age.
C. There is evidence of Conduct Disorder with onset before age 15.
D. The occurrence of antisocial behavior is not exclusively during the course of Schizophrenia or a manic episode.
In a nutshell, people diagnosed with antisocial personality disorder are indeed
antisocial (which is not the same thing as asocial, which is what people who dislike social situations are) and work only for themselves, do things that only directly benefit themselves, so on and so forth.
So to go back to your actual question, could someone who was invulnerable cope with it by being psychotic and enjoying killing things, it wouldn't so much be coping with it as already being in a state that would inhibit their ability to care about it. Would someone with ASPD be more likely to exploit their condition to kill people? Possibly, but unless doing so was a direct benefit to them there's also the chance that they wouldn't change the way they interact with others at all. Then again there's not really a case-study of an invulnerable/immortal person who also had ASPD, soooo…
I think what you've got in your mind is a more movie-like psychopath, who has a clear-cut motivation of "killing is fun, woo-hoo!" While there have been cases of mentally disturbed individuals who derived pleasure from killing, it is not the norm.
I hope I've suitably answered your question.
