Macbook turns off after a minute – not in safe mode, but in all recovery modes Problem

My friend's Macbook Pro Retina Early 2015 turns off abruptly to a black screen (as far as I can tell without logs) after booting. Takes ca. 1 minute.
The last thing he remembers doing differently is plugging in an external sound card (but it didn't crash until the next day). Supporting that, audio doesn't work in safe mode and we see no internal sound device in normal mode.

The odd thing is: We can boot into safe mode and it runs fine (except sound). However, when trying to boot into Recovery mode (on disk or online), it turns off before the mode is fully loaded.

Tests/Diagnostics

The D hardware test says the hardware is fine. 
In Safe mode, we can run First Aid and it complains "invalid btn_btree.keycount" and "fsroot tree is invalid – which it cannot fix. We could probably fix this in Recovery mode (but can't get in there) or in Single user (but this is Mojave, so doesn't exist).

When trying to boot into Online Recovery mode, it reproducibly: downloads everything, starts booting, then flickers to black once, then shows a beach ball, then turns off.

Things we tried:


resetting NVRAM, SMC
moving all Kexts in Extensions to the Desktop
plugging in the external Soundcard again


Next, our questions are:


Can we somehow fix the recovery partition from Safe Mode? Is there another way to run Disk Util (Target Disk?)
Can we somehow wipe the SSD and then restore from Time Machine?
Will creating a bootable USB stick help? Any reason to hope that this doesn't also just turn off (which seems likely to us given that Internet Recovery does)?
There is a new Mojave update which we could try installing in safe mode (but somehow we suspect that won't work/help)


And yeah, warranty is up. The disk is encrypted and has a firmware password. He has a current back-up and no money, so we'd prefer to somehow wipe the disk completely than take it in (problem is we don't know how – target disk mode?). 

Update

He took it into a shop and the guy says the logic board gets too hot to touch within 5 seconds, so must be broken. I don't know why that does't leave any shutdown logs. We are guessing safe mode works because it puts so much less stress on the logic board? We were still thinking about killing all sound kexts or something like that, because he cannot afford the replacement.
I would test booting from an external drive running a clean installed OS. If the problem persists, then it's almost certainly a hardware problem. If not, then you are in a good place to repair the issues on the internal disk.

It is odd that Safe Mode works, but Recovery doesn't. I presume this is without the audio hardware attached? What is the audio device? Does it require drivers?

It is certainly worth trying to install the Mohave update (or a Combi installer for maximum restoration of any corrupted system files).

Failing all of this, I would take it to an Apple Store. It might be out of warrantee, but they can still diagnose and fix problems. It's possible, depending on your location, that statutory consumer rights may supersede Apple's own warrantee, if it is a hardware fault.