hard drive – Question Mark on 2015 Macbook Pro — any hope saving data (and MBP)?

My 2015 Macbook Pro 13” (OS X 10.11.6) is refusing to turn on and keeps showing me a flashing question mark 😭.

I’ve followed the Apple Support instructions here, but was unable to fix the problem. I also did not have any luck with a NVRAM reset. I unfortunately do not recall the last time I backed up the computer, which is completely my fault. The laptop was working perfectly fine up until now. I hadn’t turned it on in a while and apart from running out of charge quickly it was absolutely perfect.

So far, I’ve succeeded in going through Internet Recovery, but when I attempt to “repair disk,” I get an error that says “Partition map repair failed while adjusting structures to fit current whole disk size” (see image below).

My questions are:

1. Is performing a Target Disk Mode Transfer the best chance I have to back up my data?

2. Is there anything I can do in terminal to address the repair error? I included a picture below of what came up when I typed the diskutil command. I’m not tech savy truthfully, but can follow instructions if that counts. Tried to follow this tutorial, which succeeds in resolving the disk utility repair error message by manually using fsck_hfs / sck_hfs -rc -d /dev/diskXsX. I should note that was also unable to mount MacintoshHD (option at the top of Image #2 was not clickable).

Thanks in advance for any advice or suggestions you may have!