Can you still go 160th soar with a permanent profile?

You would have to meet the physical requirements of Green Platoon so no.

As said, the answer is no. There are no exceptions. My brief stay there at the hold platoon for a couple of months before going to WOCS had me working the in-processing office. That being said, if you get hurt or have a temporary profile, they actually try to get you back up and ready (if you're willing) to safely get you off said profile. They won't bother cut you orders to get there on profile. The few that slipped through the cracks somehow were promptly put in "reassignment platoon".