A machine dual action polisher will remove most of that but it will still be visible.
If you want perfect no need to get the whole bumper done a smart repair outfit will be able to sort that at less cost.
My Brother in Law recently had his bumper resprayed as a cash job not under insurance. The whole bumper was resprayed at a proper body shop. The cost was less than £350. ( Slough) It may be worth asking around and making sure they know its not an insurance job if you want a really perfect job done.