I've not seen a LR style eyedropper. I think you'll have to do without it.
You didn't ask, but I don't think it would work well. Video files, compared to photo files, are huge. But, because it is video it does not have the depth and breadth of data in each frame. The adjustment ranges we get in video are vastly slimmer than we see in Lightroom. That means it is more important to get white balance right in the camera when shooting video.
The best tool would likely be one of the Adjustments tools, accessed by selecting the clip on your timeline and clicking on the Adjustments button on the right side of the program.
The Adjustments panel offers a number of color adjustment tools, including an Auto Color button. Adjustments can also be intuitively made by clicking on the Quick Fix Preview grid under each adjustment.