Replacement for Chicago screws?

So I have a really nice headstall I got for my mare with wide cheek pieces and brow band. The only problem is one of the reins keeps popping off the Chicago screw that attaches to the bit ring. I've already replaced it twice with new ones and no matter what it just comes off eventually. I ensured they were tightly placed but they eventually fall off.

Thankfully when it did happen, I ride with a halter and lead under my bridle and can use the lead rope. I'm wondering if latigo/leather lacing and/or snaps would be a better option. I worry, as the main attachment to the but area is also Chicago screws.

I do have alternative reins I can use for the time being. These just match the headstall.