I have put up with this problem for a long time, and here's how I remedy it...
As already stated, making a new screw is the best fix, and it's easy if you have a lathe. So I bought a $3000 lathe so I can make fifty cent parts.
Another method is to put the crew into the chuck of the drill press, and with a safe edged file, reduce the thickness of the head of the screw FROM THE INSIDE of the head. It does not take a lot of material removal to make all the difference in bringing the screw down tightly against the end of its threads in the plate.
And last, countersink the hole in the bridle t let the screw go into the bridle until it bottoms tightly against the end of its threads. You can go half way through the bridle without causing any issues, but usually don't have to. After relieving the hole for the head a bit, make a bottoming drill to match the cheese head of the screw. But I prefer to make a new screw.
Before I bought my lathe, I too used my drill press, or simply clamped my electric hand drill in my machinist's vise, and thus made a poor man's lathe. The drill press works better 'cause you can use the table to support the file on edge. Actually, use two files: one reversed, so that you take metal off both sides of the revolving steel at the same time. This helps avoid making 'out of round' screw shanks and heads.
I know what you're thinking...I paid good money and still have work to do on my lock. My reply: make a lock from scratch to see what a value you are getting in a commercial one. I concede that some locks are better than others. But don't be put off because you have to tinker...tinkering is good.