As long as all three channels should have the same value you should be able to use the WB eye dropper to set the correct value in both LR or ACR. If the point you sample is very dark then the result could be skewed by noise, so I would want to be using brightness values above around 25% in LR, which would equate to 64,64,64 in an 8 bit scale. Another issue that you can bump into with very bright values is that you may just run into clipping of one channel so it is usually good to stay under 90% or so. In this case though you do get warning that the WB cannot be set. Picking the correct colour to WB from is a matter of experience. You either include a grey card target in the fame for setting the WB or you have t know that the target you pick should be pure white/grey.
I would not be trying to reset the WB of an image using the tools in PS. If I needed to adjust the WB I would be doing that in my RAW converter, as I use Adobe tools that would either be LR (which I use) or ACR, which is provided with PS. Concurrent versions of LR/ACR both have IDENTICAL toolsets, so it doesn't really matter which one you use. I do though shoot everything in RAW. On the odd occasion that I have to deal with WB correcting RGB images such as TIFF or JPEG I would open them in either LR or ACR too. Both programs can work with RGB formats as well as with RAWs.
The channel values of a grey target can be anything you like, as long as they are all the same. 128,128,128 is only significant as it is the middle value for grey. If you use an 18% grey card for setting exposure then the values for "correct" exposure will be 128, 128, 128. It is important to remember that the "correct" exposure may not always be the right exposure to use depending on the subject, and the resulting image that you may be trying to achive.
Alan