I want us the image with the completely blown out white background, but replace its label with the one in the other image that has no glare. How best should I do this? I tried a few ways which didn't look very good. I tried to mask the layer and paint over the label with a brush, also tried using the pen tool, but I'd always leave a sliver left over and couldn't quite get it too look flawless.
Sorry, but your labels with glare are sharper than the no-glare ones. So, it's better to try to fix the originals.
You can make a selection of one label at a time, find curves adjustment which lifts up the contrast and then fix the introduced color cast with selective desaturation and lightness increase by applying Hue&Saturation in the same selection.
The color cast is an unfortunate consequence of contrast lifting in RGB images. More clever working, for ex. visiting in the Lab color mode would remove the color boost problem. As well one could use blending modes.
The edges can become dark, but that can be fixed by pushing white against the selection border with the smudge tool. An example:
The selection is made with the quick selection tool. This curve removes a great part of the reflection, but generates some yellow dirt. It's fixed by opening the Hue&Saturation adjustment and desaturating+making lighter a hue range near yellow. All settings are essential!
The corrosive attention was selected with the polygonal lasso. The best curve was different and it left a faint reflection + some yellow dirt. The reflection remnant was dark red.
The edges got unwanted black stripes - maybe a bad selection or they were already dark. Keep the selection and push with the smudge tool wite fill color against the selection border. This is the result:
Not asked (like the whole answer):
Get a tripod. Then you can take overall sharp images. To get them you must use manually adjusted small aperture and long exposure. There's no glares if you have no bright light behind the camera.The light should become elsewhere than behind the camera! Prepare the light for it! Get some photograph technique guidance. If you have a tripod it's no problem if the small aperture and the not so bright light make the exposure long.
Use so high optical (not a digital one!) zoom that you get the subject grabbed in as big size as there's room in the image. I know that you cannot go too close because the perspective becomes exaggerated.
Not sure I understand.. copy, paste, color correct. Is that not working for you?
The labels themselves are very easy to copy because you really don't have to be all that accurate with any selection. They are white labels with white at the edges, so merely select the interior of the labels, no need to try and carefully match the outer edges. I just used the Polygonal Lasso to quickly grab the inner area of each label (separately) and copied.
Color correcting can be a challenge due to the overly dark labels in the second image. I had to split the top label into two parts so I could correct each half separately. And this was quick.. much more time could be expended on correcting the tones.
Note that the top image is bad in different ways in both your photos. If you are seeking absolutely crisp appearance, the optimum way to achieve that would be to retype the top label reconstructing its contents entirely. Which is what I would ultimately do if the image were destined for reproduction.
I'd also probably reconstruct the Corrosive label as well. Especially since that artwork can probably be found in a vector format for free as it appears here. It's a fairly standard label.