Wood – How strong is a lag bolt through the narrow face of a 2×4 stud (meaning it only has 1.5″ to grab)

You can overkill the job as much as you like, but it's easy enough to use a studfinder or a magnet (looking for screw-heads) to establish the location of the stud non-desctructively, or drill some small holes that will be covered by the rack (and which you can spackle) to destructively identify for certain where the edge of the stud is, and thus infer the center. Given an inch and a half of surface, you don't have to be too precise not to miss the stud.

I've mounted dozens of wall-mount racks and never bothered with more than a studfinder and/or magnet, and not one has fallen off the wall.

If you use long enough lags to get a good bite into the stud, your first proposed method should be fine. If the "lever" action is high enough to snap a lag bolt, then it will be strong enough to snap the 2x3 as well.

From a strength perspective, it doesn't matter how long the lag bolt is - the torque and shear at the junction of the wall/bracket interface is all that matters. It doesn't matter if the lag bolt sticks out 3" from the wall, or if just the bracket sticks out 3" from the wall. You get the same torque applied from the same downward force if you assume the lag/bracket/mantel assembly is one rigid body. And if it's not, then something internal in the shelf has already failed, and you don't need to worry about the torque loading on the lag at the wall.

If you wanted to use shorter lags for some reason, you could always countersink them by drilling into the 2x3 with a bit larger than the lag head.