Yes, you can use your JEE Main score to apply to private engineering colleges in Karnataka. But home state quota is usually available only in government colleges, not private ones.

Many good private colleges in Karnataka accept JEE scores, like:

1.RV College, Bangalore

2.CMRIT

3.Dayananda Sagar College

4.New Horizon College

Apply through KEA counseling or directly to the college. Check each college’s website for details a nd deadlines.