The thing to keep in mind is that all of these third party products are reverse engineered and probably haven't been tested in every conceivable scenario before being released to the public. The cheapest ones and the ones that first appear on the market after a new model is introduced are often the worst. Some of the more reputable third party brands tend to take a little longer to get their accessories for new camera models to market. For third party grips I've found that the 'Pixel Enterprise Limited' Vertax grips tend to be the most reliable and trouble free. 'Zeikos' and 'Meike' have been pretty good. I've never had a 'Neewer' grip but other 'Neewer' products I've used tend to be hit or miss at times.
One thing you might try is to always place a genuine Nikon battery in the right bay in the grip. From an answer at amazon to a question regarding the grip linked in this question:
(It should work) as long as you install the real Nikon battery first on the right side, then any other battery. Using a non-Nikon battery without the Nikon battery first will throw an error. To clear the error remove the battery then power on with no batteries in the unit to clear any remaining charge.
Another answer on the product page at amazon in response to a question regarding fit:
Overall, the grip fits the camera well and makes it feel a lot more sturdy. The only con; I had early issues with the grip keeping connection with the camera. I fixed this, however, with a couple of small adhesive pads attached to the grip between the camera and the grip, on the opposite end of the battery insert. Now, when attached and tightened to the camera, the battery insert portion of the grip is forced down and sits snug without losing connection.
If you want absolute rock solid dependability in a battery grip you'd probably have to break down and buy the Nikon branded grip - if such a grip existed. But one doesn't exist for the D5200.
For other cameras that do have available grips offered by the manufacturer, that's the best solution for 100% functionality and the highest level of reliability. Sure, they are outrageously priced. But the only other options are all too cheap because the competition for third party products is all about price and almost nothing else but price. It's a shame there are no third party grip makers that make high quality grips at a reasonable price the same way the best third party battery makers make batteries that sometimes outperform the OEM batteries for 1/2 to 1/3 the price.
I once had a cheap third party grip for a Canon 7D Mark II that did pretty much the same thing as yours is doing. It would only do it occasionally, but it always seemed to be when in the middle of shooting a burst of action frames. Of course the instant power-down meant I lost all of the images in the buffer!
I finally figured out that at certain shooting positions the grip was flexing just enough to lose contact with the camera at the connection point inside the top of the battery well. In effect, it was like a computer that was being unplugged without being properly shut down first (and without an UPS). Once I figured out what the problem was I could easily reproduce it by flexing the grip in the wrong way. In defense of the grip in question, this only started occurring after the camera with grip attached had taken a pretty nasty fall from vertical to the ground while attached to a monopod extended at about 60 inches.
I've had another third party grip that would act flaky at times. It caused a control on the camera's body to not function. Rebooting the camera solved the problem until the next time it decided to do it.