A lunchbox locker in 2wd with the axle disconnected will behave basically like an open differential.
In AWD with the axle connected, it will behave like a lunchbox locker. Decent on the trails but "less than ideal" on road. When you apply power (i.e. push on the gas) the locker will engage and not let the wheels run at different speed. So you get "lock steer", which seriously impacts turning radius because the front tires can't turn at different speeds. Coasting around a corner will let the locker ratchet. But if you're on very low traction roads (snow/ice) you will lose traction easier. Getting in and out of the throttle going around corners will take some getting used to.
In AWD with the axle disconnected, you'll essentially be running in 3wd. Because one of the shafts is free to spin any speed, it will do so.
The only time I ever ran an automatic locker (lunchbox, full case Detroit, etc.) in a front axle was when that axle had locking hubs. When I hit the street, I'd get out and unlock the hubs so I was in true 2WD. I personally wouldn't drive a vehicle with an automatic locker up front unless I had that ability.
But one other thing to think about. The shafts have a retaining clip on them in the Eagle differential. Read up on the Eaton E-Locker thread to see what he's having to do to be able to put the clips in. I would be willing to bet a lot that a lunchbox locker wouldn't be easy to set up to let you use those clips. It's worth investigating but it may not be possible to do what you want to do.