Can lettuce wraps be less messy?
Not sure if this is what you're looking for, but rather than trying to solve the problem, you can also just accept it. Bring it all to the table, and just make wraps as you eat them. It won't make it less messy, but it won't be much slower than eating pre-made wraps, and you'll be spending that time at the table with people instead of waiting to eat.

To reduce the mess at least a bit, go for leafier lettuce, avoiding the parts with thick, stiff ribs, and definitely avoiding varieties like iceberg. (I've seen people recommend iceberg for wraps, but it seems like an awful choice to me.)

You can also potentially use other greens. There's napa cabbage and blanched cabbage, but also things with a bit more flavor that you might find in an Asian grocery like perilla (shiso, kaenip, ...) - I'm not sure I've had all of them, but the leaves used for ssam are probably good ideas.