Machine Check Errors are usually caused by a hardware issue. Make sure to disconnect all external devices in case one of them is causing a problem.
Either the new SSD is bad, or the laptop has some other hardware issue (perhaps those other hardware issues are why you thought you had to replace the original Apple SSD, although the original Apple SSDs for this laptop do tend to have a higher rate of failure). I should also mention that Apple does have a free repair program for some faulty SSDs in this particular model laptop:
https://support.apple.com/13-inch-macbook-pro-solid-state-drive-service
As well as a free Battery repair program (only affects certain laptops and only an AASP can confirm if the laptop qualifies):
https://support.apple.com/13inch-macbookpro-battery-replacement
FYI, here is the current list of publicly acknowledged free repair programs for various Apple products (some products such as your laptop may have multiple programs):
https://support.apple.com/service-programs
I suggest reinstalling the original Apple SSD and having Apple or an Apple Authorized Service Provider examine the laptop to see what they find wrong with it. If you go to an AASP, I suggest taking your OWC along as well as an AASP may be willing to test the laptop with the OWC SSD (Apple won't have anything to do with third party parts, but some AASPs might). Take your charger and charging cable along as well just in case they need to be checked out as well for the possible charging issue.
If you have a USB drive available you could try installing macOS to the USB drive to see how the laptop works. You will need to have a working internal SSD installed though since the macOS installer requires it (you may want to try with the original Apple SSD installed in case it works just enough to allow the macOS installer to work). If the laptop works Ok while booted to the USB drive, then it may indicate the issue is with the OWC SSD. However, if the laptop crashes even while booted to the USB drive, then it doesn't really help us unless the crash log provides some more/other information. If you have access to another Mac capable of running macOS 10.13+, then use that Mac to install macOS to a USB drive so you can test the laptop without an internal SSD installed. This latter option may provide the most information about the issue.
Troubleshooting this is not easy because of how the Apple hardware and macOS work. The USB-C Macs are the hardest computers to troubleshoot.
Good luck.