2002 - Nebraska vs Miami. Nebraska was not in the Big 10 at this point.
2003 - Washington State vs OU. OU is not in the Pac 12 or Big 10.
2005 - Texas vs Michigan. Texas is not in the Big 10 or Pac 12.
2006 - USC vs Texas. Texas is not in the Big 10 or Pas 12.
2011 - TCU vs Wisconsin. TCU was in the Mountain West that year.
2015 - Oregon vs Florida State. Florida State was not in the Big 10 or Pac 12
2018 - OU vs Georgia. Neither team is in the Big 10 or Pac 12
There is no reason why a non Pac 12 or Big 10 team cannot play in the Rose Bowl.
Regarding restrictions in California, they are not going to budge and I can honestly say I don't blame them.