Back up MacBook without display
The simplest method (assuming you have or can borrow/rent one) is plug in an external monitor. That may not be possible or practical but it has to be said...

The problem here is knowing if the Mac is on or off. You can check in a darkened room to see if you see ANY light change (keyboard, display) when hitting the power button. Also in a quiet room listen for fans or drives spinning up (assuming you have spinning drives and not an SSD) And often turning on a Mac will make some noise when it is powered on.

If you can get to the battery and take it our or disconnect it readily that would confirm it is indeed off.

You can usually shut down any computer (Mac or PC) by holding down the power button for 15 seconds or so. Once you do that hitting the power button will power it up.

Forgoing all of that you could just plug your Time Machine drive into the Mac and watch for the drive activity light. More than a 30 seconds or so of drive activity might be a good indicator that macOS was accessing the drive. Let it run overnight and then plug it onto another computer and check the date stamps on the sparsebundle and/or its contents to see if it was modified overnight.

There are likely other ways that I haven't thought of but those should get you started, unless you've already tried all of that...