Precision 7540 will only boot Linux if charger is connected

I recently got a new Precision 7540 laptop for work. The first thing I did was wipe the SSD and install the latest Linux Mint. It runs great, with one caveat. It will only boot if the AC adapter is plugged in. On battery, 9 times out of 10, it won't cold boot. I see the Dell splash screen come and go. I see a message about booting in insecure mode because secure boot is disabled, and that's it. I never get prompted for my disk decryption key, although the keyboard backlight activates if I touch the keyboard. I have to poweroff by holding the power button to get out of this state. If I plug the charger back in and power on again, it comes right back up. Once I've made it to the login screen, I can unplug the charger and everything keeps working normally from there. I've reinstalled without full disk encryption and it doesn't seem to make a difference. I've tried the latest Ubuntu, Kubuntu, Fedora, as well as Ubuntu 18.04.3, all with the same startup issue. The issue is there before I install Nvidia drivers and afterwards, so that seems to have no effect. Ubuntu 18.04 seems to get farther; I actually see the Ubuntu logo with the 5 white dots that turn orange, and then back to white, and then back to orange... for 20 minutes until I hold down the power button, plug in the charger, and power on again. And then it comes right back up!

I spoke with Dell technical support and their stance was that since the laptop shipped with Windows, they can't provide support. So I reinstalled Windows 10, and frustratingly, the problem isn't there. I was hoping it was an obvious hardware problem that could be repaired.

I know Dell ships this model with Ubuntu 18.04 installed, so I know the hardware 'should' be able to work in that configuration. However, Dell has their own customized Ubuntu image. Because my system shipped with Windows, I don't have access to an official Dell Ubuntu image that would likely contain any needed patches/fixes, if the issue truly is limited to software.

My main concern is that I'm coming up on 30 days with this laptop. If I can't gain confidence that it's going to work in the long term, I'd rather send it back and eat the cost of restocking rather than risk being stuck with a laptop that's not fit for purpose. Is anyone out there running a stock (non Dell customized) version of the latest Ubuntu on the Precision 7540? If so, do you see this same issue?