Perhaps you might want to consult a sports doctor. Search LYN. Dr Asron sports clinic or something.

Push ups. And close grip/triangle/diamond push ups.

It's a good exercise that helps to strengthen the shoulders and the muscles around it (including the rotator cuff and those whachamacallit muscles.)

What's wrong with Arnold press?
Form over weight. If tak boleh tahan, drop the weight and get good form.

And yeah, anything that goes to the back of the neck carries high risk of injury for the shoulders. Lat pulldown to back & overhead press from the back. Not recommended until one strengthens his rotator cuff muscles.