APFS: fsroot tree is invalid after TimeMachine backup - how to recover and avoid in the future?

System

MacBook Pro, late 2013, 1GB SSD (brandnew, recently replaced by Apple), APFS (no journaling, case insensitive), High Sierra 10.13.2, TimeMachine to network HDD.

What Happened

• Mac stopped working,

  no space left on device

  .
• Reboot failed.
• Tried to boot into recovery mode with Command-R and run First Aid from Disk Utility - failed, because apparently the recovery system also resides on the same disk which seems to makes fsck on APFS impossible.
• Tried to manually delete some files via

  rm

  , got

  no space left on device

• Tried to truncate some files manually via

  cat /dev/null > somefile

  , got

  no space left on device

• Booted into recovery mode with Shift-Command-R (downloads the system from Internet) and ran First Aid again. This time with limited success:

** Checking volume.

** Checking the container superblock.

** Checking the EFI jumpstart record.

** Checking the space manager.

** Checking the object map.

** Checking the APFS volume superblock.

** Checking the object map. error: invalid dstream.size (10730881024), is greater than dstream.alloced_size (71151616) error: xf : INO_EXT_TYPE_DSTREAM : invalid dstream error: inode_val: object (oid 0x16309a1): invalid xfields

** Checking the fsroot tree. fsroot tree is invalid.

** The volume /dev/rdisk2s1 could not be verified completely.

So apparently the fsroot tree is invalid. I've searched, but wasn't able to find any usable advice on how to fix this (except of course, reformat and restore from backup, which I'd like to avoid).

Additional Background Info

On the system is a Parallels Windows VM with a virtual 100GB harddrive (yes, one big file), which was recently used (so a backup was needed). The last time I have used the computer, roughly 20GB were still free on the macOS SSD. For a day or so, TimeMachine backups have not completed, but no error message was shown. When the problem happened, I had left the machine turned on over night to complete an incremental TimeMachine backup. The connection here is, that TimeMachine is apparently using APFS snapshots. I suspect this is the root cause of why this mess happened.

Questions

1. Is there a way to fix this (without reformat and restore from backup)?
2. What's the best way to avoid this in the future (especially with regard to TimeMachine)?

Thanks.

Same question on Apple Stackexchange: https://apple.stackexchange.com/q/311263/62735