Keep flashlight on while in camera
This seems like a device specific issue. Some phones will allow the flashlight to be on while the camera is on, others won't.

The phone I'm using, Galaxy S5 has no issue with the flashlight that comes preinstalled, being on while the camera is running.

So if you're anyway buying a phone for this, Galaxy s5 or any other phone that allows it, should be your choice. (In my humble opinion, instead of buying this phone, you're better off buying a security camera with night vision...)

In the screenshot below, I have the manything app in recording mode, and the flashlight on, and I'm facing a mirror. 



Using galaxy S5 on AT&amp;T Android 6.0 flashlight widget that comes with the phone.

Note: other flashlight apps didn't work while camera is on.

I wouldn't post this as an answer had there not been a comment from OP saying: 


  @Abochur I mean I'm literally going to buy an S5 for this express
  purpose :). Could you help me make sure I get one that'll let me use
  the flash like you can?