You don’t build a wall across the lake. You wall off the lake with controlled access points for legal access to the lake. Really simple.
Have you ever been to that part of the state? The terrain is very rugged with huge canyons. A wall is just not practical. People don’t tend to cross there anyway because there are few towns nearby and without water, they are goners.
But unless those access points are manned 24x7 then couldn't the illegals sneak in through them?