MacBook Pro shuts down when starting from the (not fully charged) battery

Did you calibrate the battery as instructed by NuPower? This involves charging the battery to 100% and keeping the charger on it for at least two hours. Then disconnect the charger and use the laptop until the laptop powers off when the battery reaches 0%. Reconnect the power adapter and let it charge to 100% once more.

Even though OWC, a respected supplier of third party Apple accessories & parts, sells NuPower batteries....over the years I have had a lot of these batteries fail prematurely. Unfortunately the quality of third party Lithium batteries is extremely poor even when purchased from a respected vendor (yes B&H is a respected vendor).

Try removing the battery and only using the power adapter to see how the laptop works without the battery. This eliminates the battery from the equation and hopefully the charging circuitry on the Logic Board.

Most times I would not expect a battery to cause such an issue when using the power adapter....especially if you have tried two different batteries. I suspect some other sort of hardware issue. I'm not completely ruling out the battery or battery related issue like an issue with the charging circuitry on the Logic Board. You usually want to start disconnecting things to discover what if anything may be causing the problem.

It also helps to boot from known good external media to quickly eliminate any issues with the OS and installed software on the main internal boot drive and also any possible issues with the internal SSD. Unfortunately a 2009 model does not have access to Internet Recovery Mode (unless it is possibly a Late-2009 model running macOS 10.12.6+). If you have a spare external USB drive, then you could try installing macOS to the external drive and see how the laptop behaves while booted to the external USB drive containing a full macOS installation (avoid installing third party apps or restoring from backup).

You can try booting into Safe Mode to see if that makes any difference. If Safe Mode works fine, then it may indicate an issue with some third party software, or it may indicate a GPU issue as I alluded to previously.

Remove the battery and see how the laptop works with just the power adapter connected. Most likely the battery is at fault since a battery, especially a third party battery is the weakest link. Of course there may be some other hardware issue with the laptop. You can try running the Apple Diagnostics to see if any issues are detected although you will more than likely need to disregard any battery issues since many third party batteries will cause a diagnostic error even when the battery is brand new & healthy.

I also have to consider the possibility the battery charge level has nothing to do with this problem. The 15" & 17" models tend to have a lot of GPU failures which can show up other than just graphical anomalies on the screen. The laptop should boot just fine without the battery being installed.