I've tried the following:
Both seem to take my Linux VMs offline on a HyperV Box. We used Rsync before and we are considering going back to that for the Linux VMs.
I'd like to know if there is any backup software that works well with HyperV Linux VMs and does not take them offline or pause them when doing a backup? Also must not be too pricey
You should be able to backup those VMs without taking them offline...using that software.
Make sure Hyper-V VSS Writer is functioning.
And you might need this: https://www.microsoft.com/en-us/download/details.aspx?id=29991