Allow me to state the obvious...

1) Per company policy, your current employer is on schedule to start your GC process in about half a year.
2) Make 1) known to your prospective employer, and politely ask if it could promise as much (i.e., start your GC process within half a year); in doing so, make it clear that your service/availability depends on it.
3) If your prospective employer says yes, again, politely ask that it put that promise in writing.
4) If your prospective employer says no, stay put, and follow up with your current employer on its policy promise.