It appears you have the MacBook Pro 15" (mid 2010) model laptop. This model is known to have GPU failures, however, I have seen some of the GPU failures manifest as dead logic boards which Apple replaced for free GPU repair program (now expired). If the laptop is running Ok without the battery, then this is not likely the problem.
From the Kernel Panic report I see two possibilities. One item appears like it might be an NVRAM read error. You can try resetting the PRAM using the instructions in this Apple article (just make sure to hold the reset for at least three chimes):
Reset NVRAM or PRAM on your Mac - Apple Support
You can also clear the NVRAM using the command line:
sudo nvram -c
This will prompt you for your admin password, but nothing will show on the screen as you type the password so press the "Return" key to submit the password. You need to reboot about issuing the command to load the default NVRAM settings. The traditional PRAM Reset seems more reliable though, but it does not hurt to try both.
The other potential item shown in the Kernel Panic log is the Virtual Box drivers were the last ones loaded, so maybe there is an issue with Virtual Box.
I don't believe either of these items in the Kernel Panic would affect the BIL. The only time I have ever had to replace the BIL was when the cable was accidentally torn, or when there was liquid damage to it or the connector on the Logic Board. IIRC, the BIL cable just snaps down onto the Logic Board connector for that model, but if it is instead a flex cable sliding into the connector on the Logic Board, then make sure the cable is fully seated & the connector latch is down.
How are you performing the SMC Reset? I believe I used to use the keyboard combination of Left Shift + Left Control + Left Option + Power Button (even if Apple's instructions today don't mention this option). I personally like to Option Boot the laptop and let it sit at the Apple boot picker menu when I perform the SMC Reset as it will immediately power off the laptop when the key combination is pressed.
Assuming the BIL part is not damaged and the cable is fully seated, I would expect the Battery to be the reason followed by a bad Logic Board. I guess you could purchase a BIL to test as it should not be expensive (you can just connect it without fully installing it so you don't risk disturbing anything else). If you still have the original Apple battery, then see how things behave with the original Apple battery. The fact there may be an issue reading the NVRAM may indicate some sort of Logic Board issue especially if resetting the PRAM does not eliminate that error in any later Kernel Panics. I'm assuming the "NVRM" mentioned in the Kernel Panic log means "NVRAM".
You would not hurt the laptop by holding the power button without the battery. The only time I ever had an issue with a Mac was if I held the power button down while connecting power to the Mac, but I believe the issue was only temporary (I no longer remember the exact details).