I can't help but think the major problem with Hawk is his not knowing if and when he is going to get grabbed. He does not trust that you will trust and respect his space and comfort. I currently have 26 lovebirds and have had lovies going on 11 years, and know for certain that most tame and untame lovies will become and continue to be jumpy when they think they may be grabbed at some point. This is a major reason I think you should avoid touching or grabbing Hawk at all costs. In my mind, the only time you should put your hands on Hawk til' he builds trust is if there is a medical condition or immediate danger. If he flails to the ground, this is the perfect time to get a stick and gently work with him on stepping up on it to get him back on his playgym. Use voice commands, a gentle voice, and be totally patient. If Hawk starts getting upset, then give him a few minutes to calm down and try again. I would also talk to Hawk in a soft voice like he is a person, and let him know what you are going to do. I also suggest you continue to leave his cage door open and let him continue to come out and be on his playgym. Who cares if he doesn't play with toys. He will do what he wants and that's what counts. I guarantee if you stop touching him and give him time, he will learn that you are not going to grab him and he will become more comfortable and let you get closer.
I would definately not get another bird right now. It sounds like Hawk is a work in progress and needs your time and attention. Once you guys build trust, then I think putting some thought into another lovie is a good idea. Right now, another bird would be too much and take away the time Hawk needs with you.
BTW, Hawk is beautiful. He's a slate masked lovie. I have a slate masked Named Olive who has a peachface Creamino as a mate.