#5 or #6 screws seems awfully small and light and if it was me I'd want the adjusting screws to be larger. More information would be helpful.
How deep is the recess in the chuck body?
How wide is the back plate spigot (aka, shoulder, register, nose)?
The adjusting screws don't necessarily have to be centered on the width of the recess in the chuck body, or the back plate spigot, they only have to contact the spigot somewhere on its width with enough metal underneath it to make flat surface contact and not break the metal. Sometimes, unintentionally, when you are trying to snatch that last .0005" of adjustment, a lot of force ends up on one or more of the screws and if there isn't enough metal under them they could break a chip off.
In this case, if the spigot is 1/4" wide the contact point (center of the screw) could be moved out to say 3/16" from the back plate face. If you allowed say 1/16" clearance (from the nearest thread crest) from the back plate face that would easily allow a 1/4" adjusting screw. I would then work that back to the chuck to see if a 1/4" screw would fit the metal available in the body. There should be a balance to be struck between largest screw size and the minimum amount of metal surrounding the screw threads.
As for lock washers or Loctite, I have two such chucks both of which bolt on from the front (with Allen head alloy bolts) and neither of them came with or call for lock washers. I think you overestimate the potential for those screws to loosen or back out. I'm not saying it can't happen, but I've never had one loosen up (in 40 years.)
