Will a ss transmission fit a 2007 2.2l n/a engine

I have a 2007 Chevy cobalt 2.2l with a 5
speed manual transmission with a short shift
kit(I’m not sure what brand it is I got it with
purchase) and it’s been grinding into 2nd first
it started with just down shifts at high rpm
then just normal up shifts at low rpms and
now when I shift into 2nd and go above 2500
rpm it pops into nuetral I think the trans has
200xxx bc that’s what’s on the odometer but
the engines swapped with 160kms and had a
recent clutch put in