Rationality does not in any way depend on the PSR. It is quite rational, and not uncommon among philosophers and scientists alike.
I've heard some philosophers almost hint that not accepting the PSR is irrational. I don't know if I would agree with this. Certainly the PSR, if sound, is not something that can be known in an analytic a priori fashion; in that sense the PSR is not self-evident.
There is no rational reason to accept the PSR, and the best reason to reject it is simoly that thrre are lots of things for which we still lack exlanations.Given that it is proposed as an axiom, a rule of thought, I would recommend rejecting it, just because it is useless in that role, but it's not irrational to reject or accept it.