Steps to reproduce:
1. Open "case 804252 repro" project
2. Open "lightProbe test" scene
3. Clear GI cache in Edit>Preferences> GI cache and bake lighting
4. Change "Directional light" object light intensity higher than 1 (I suggest 5-8)
5. Re-bake lighting

Results: Light probes have way too dark baked light with bright spot in the shadow area (screenshots in last edit).

Note: Prerequisite - disable Precomputed Realtime GI in Lighting settings.

Reproduced with: 5.4.0b21, 5.4.0b22, 5.3.5p3
Not reproducible with: 5.4.0b20, 5.3.5p2

Resolution:
Hi Paul! The fixed light probes are now matching Unity 4.x behavior. I have recreated your scene and baked it using Unity 4.7.2.

Unity 4.7.2 gamma color space:
Unity_2016-06-17_10-20-58.png

Unity 4.7.2 linear color space:
Unity_2016-06-17_10-26-25.png

The bright spot is caused by the intense direct lighting causing a ringing artefact on the opposite side. This is an inherent limitation of spherical harmonics. We can't fix this easily without going for a different reconstruction algorithm.

You can try the fancy L1 reconstruction code that I posted here http://forum.unity3d.com/threads/unity-5-3-1p2-light-probe-issue-dark-spot-in-the-opposite-direction-of-the-directional-light.378516/#post-2538912 and perhaps you can get rid of most of the ringing but the shading cost is much higher so it might not be suitable for all platforms. Keep in mind that the light probe SH axes is now swapped around so you have to swizzle them in the shader.

We might implement the fancy L1 reconstruction code into Unity at some point, if the demand is there (vote on feedback if you need this).