Hello,
No can do, .
If we talk on the basis of of rules or conditions, so one cannot change his/ her stream in class 12th if you are going to change your board then also, because your 11th class result will also be considered there, but  if you are going through the migration then it is possible, and again you can also do if you are going to private in class 12th from any other board.
Moreover i would like to suggest please do not change the board unless and until you have no option left.
Good luck!