If your in class G your on your own.
ATC doesn't release the Jet into class G it stops giving it a service in controlled airspace when it exits. Per say there is nothing they can do to prevent the Jet going into class G. A lot of operators have SOp's to prevent crew going into class G unless there is no other option. Biz jets tend not to have that SOp because its all about minimising costs and they do it regularly.
Most IFR jet pilots unless they have done GA in the area will not have a clue where the high intensity training areas are.
The issue you have brought up is a airspace classification issue.
There is no fix for it in Class G.