I rarely use the dropper anymore, because it doesn't usually give me the results I want (accurate reproduction of all colors in the photo, even when shot under less than ideal lighting). The same goes for Auto white balance as the default WB when opening a raw file. If you're not shooting raw files, then you've got to get the WB very close when shooting or it's pretty much a lost cause to do any batch processing unless the light is very controlled and consistent in every frame. You're not going to be able to make large changes to the color bias of a JPEG image without reducing the overall quality of the color of the image in terms of both available colors and the available saturation levels of those colors. This is because the raw to jpeg conversion process throws away all of the additional information you need to shift the color bias.
Selecting a color temperature (which moves along the blue - amber axis) and also adjusting along the magenta ←→ green axis is the best way I have found to get a good standardized color. By hovering your cursor over a point in a photo that is supposed to be a neutrally colored object and observing the (R,G,B) values for that spot you can see where your color bias is currently set at and correct it. Once you've found the right combination to render that neutrally colored spot as a numerically neutral color where the (R,G,B) values are all the same or very near the same then you can batch apply that same WB to the entire batch. While this won't give you perfect color in every frame if the light has changed a little, it should get you fairly close and give the various frames a consistent look while also reflecting the changes in the light as you shot.
This method also allows you to to apply a warm or cool, or even magenta or green tint (very useful to get skin tones right) in every frame at the same amount. For instance, if you are shooting at the golden hour shortly before sundown you probably want the amber and other warm tones to come through in the photos. By using the numerical (R,G,B) values of a neutrally colored object to set your WB, you can leave the same exact bias in each frame.
Let's say you want the light from a session to appear neutrally white. The color temperature is set to 5200K with no bias along the M ←→ G axis. You hover your cursor over a neutral gray object in the frame and the the (R,G,B) value is (185,205,235). The picture is probably obviously way too blue to your eye just by looking at it. You would need to adjust the color temperature to the right until you find the number that makes the red and blue numbers fairly close. Let's say at 5900K the same spot now has a value of (210,198,205). That's closer, but now you can probably see a magenta tint in the image, so you move that slider a bit to the green side until the numbers are (208,208,208). By pulling a little magenta out of the WB you also decreased the red component slightly and increasing the green bias also increased the blue just a bit. Sometimes you have to go back and forth adjusting the color temperature (or the blue ←→ amber fine tune control) and the magenta ←→ green axis to dial it in. I usually will sample two or three areas of neutrally colored objects. With shots taken in daylight conditions, those areas in shadow will probably have a bit of bias towards blue. The brightest areas (that are not fully saturated in any of the three color channels - using those won't be useful because even if there is twice as much red as blue in the scene both will have the same number: 255) might have a bit of a red bias. Shots taken under artificial lights can show just the opposite: shadows go a little towards red while highlights are bluer. I've found the green levels can vary wildly under outdoor stadium lighting. Some types of lights have almost no green component while others have a ton of it.
Once you've found your image's neutral point, it's then a simple matter to move the color temperature a little to the left or right to warm or cool the overall look of the photo. You can also do the same along the magenta ←→ green axis. And since you're using numerical color temperature and white balance instead of auto or the dropper tool, it's very easy to apply the same offset to every image.
Then you have specific situations where batch processing isn't very practical at all. Example: The problems with stadium lights are compounded by the way they cycle with the alternating current running through them. The overall brightness can vary by as much as a couple of stops from the peak to the valley of each 120hz cycle. The color can also shift drastically and be very blue/white and fuller spectrum at the peak, and very red/brown and much narrower spectrum at the trough. Unless you have one of the newer Canon bodies with the flicker reduction feature which times the release of the shutter with a peak in the light cycle, you probably won't have much success batch processing images taken in such conditions. If you're shooting at shutter speeds shorter than about 1/125 second you'll even have more than a few images where the color changes significantly from the top to the bottom of the frame. The faster your shutter speed, the more pronounced this effect will be. If you're shooting wide enough, you'll also catch the light from two different towers that may be out of phase with each other, so one is dim and brown while the other is bright and white and and instant later (as in the time it took the slit between the shutter curtains to traverse the sensor) they've reversed and the first is now bright and white while the second is dim and brown.
Addendum based on comments:
One thing you might also try is to apply the white balance tools in such a way that it is as neutral as possible. Then add the warmth you desire to each image using selective color to boost yellow and orange and to a lesser degree red and green. Lightroom gives you separate control over the hue, saturation, and luminance of each color range. The HSL correction would be applied to each image independent of the white balance setting.
Another thing that applies equally to each image independently of the selected color temperature and WB settings would be the camera calibration for red, green, and blue. You could build a profile there that pushes red a little, pulls blue a little and then adjust green to suit.
A note about LR vs. other applications:
One reason I prefer to use Canon's Digital Photo Professional 4 for raw conversion is the very fine level of adjustment it allows for many parameters. Brightness can be adjusted in 0.01 stop increments. Color temperature can be adjusted by increments of 10 Kelvin (i.e. 5200K, 5210K, 5220K, etc). The fine adjustments available for the blue-amber and magenta-green axes are equally precise. So are the adjustments for gamma (three separate controls in 0.01 stop increments for floor, midpoint, and ceiling), contrast, highlights, shadows, saturation, unsharp mask, selective color, lens correction (peripheral illumination and distortion correction can be applied from 0-100% in 0.1% increments, CA correction control is equally precise), cropping/rotation (widths/heights can be adjusted in increments of a single pixel and rotations can be applied in increments of 0.01º), selective color (HSL), etc.
I'm pretty sure, though that LR also allows finer adjustment of many parameters than the default amount of movement caused by each press of the arrow keys on your keyboard. (Unfortunately, LR is not installed on the computer at which I'm presently sitting.)