Found a setting that seems to have solved the issue for me.
Windows 10 (and some older versions) has hidden Power Options under the following registry key:
HKEY_Local_Machine\System\CurrentControlSet\Control\Power\PowerSettings
Under there, you can see a bunch of different keys. They are organized in the same manner as the settings you see in "Change Advanced Power Settings" under an individual Power Plan's settings. I found a key with the description "Power buttons settings and configurations", and under that was another key (833a6b62-dfa4-46d1-82f8-e09e34d029d6) which had the description "Enable forced shutdown for button and lid actions".
There is two options from here. If you want to make this option appear in Power Options -> Advanced Settings, you need to change the "Attributes" REG_DWORD value to 2 instead of 1. If you want to change whether the setting is enabled or not for your power plans, you need go to each key under DefaultPowerSchemes and flip the values AcSettingIndex (plugged in) and DcSettingIndex (not plugged in) to 1 instead of 0.
I've included a reference to MSDN here, which should help explain it better.
If anyone sees an issue with using this setting, or if anyone has a better idea, I'm all ears. For now, this seems to get around the special settings that apply on the lock screen, but I have no idea why the lock screen is set to behave differently than a logged in user anyways.