Sounds like you’re on the right track. Take a look at the e2fsck manpage, in particular the -b option.
Never run filesystem utilities on (or worse: mount) VM filesystems in dom0. The filesystem on-disk format itself, and not just the files stored in it, is a complex data structure that’s fully controlled by the VM, so it’s best treated as potentially malicious. Filesystem utilities (or kernel filesystem code) can’t be assumed to be robust against this.
(It’s possible to attach the private volume to another VM in case the corruption really turns out to be severe and you need to resort to more advanced data recovery software like TestDisk.)