You should really look into a chicken sling. You could probably make it out of the basket you are already using. If he's wanting to stand it will help him a lot.
You could add support for his neck too. Have the sling sit higher up on his chest like. I've seen slings made from an old towel with holes cut in it and some binder clips.
You are keeping a very good eye on him. See how he does in it and go from there.
Hello! I think you are doing a wonderful job with taking care of your rooster. Every bird deserves a caretaker as dedicated and caring as you are. It's never fun losing our feathered friends. I lost my favorite chicken to egg yolk peritonitis a couple days ago and I wish I could have done more for her. But you've already gone above and beyond for Brownie. I'm impressed! And very happy to see the progress he's making. Keep up the awesome work. I'm rooting for you and your roo!