Dry wall screw was the first thing that came to mind, short one (thankfully) and I'm speculating that lump maybe the head. As someone said, it stuck into her when she laid on it, went in side ways and came out.
I agreed cutting it at the skin but then reading through the whole thread, good points were made about why it may not be a good idea. It's hard to figure what to do, we're all not there to see and we all have different levels of experience in "surgery".
In the last picture, there's some brown or reddish brown stuff on the screw towards the skin. Is it rust or something from inside? The more I look at the pics, think I'd get someone to help, load up on antibiotic and disinfectant, make a small incision on the skin a to get it out. Super glue the incision close or some fast stitching.
Is there anything that can be given the chicken for pain? Baby Aspirin? I'm no expert and I feel for the OP, having to decide which way to go.