I understand the requirements here, but unfortunately it is not easy for "alternative" virtualization platforms to deal with Microsoft's shenanigans. If you think that even Citrix gave up on quiescent snapshots in xenserver, you understand the challenge here.
I guess you could try calling vssadmin create shadow via ssh or powershell, but that I have never tried actually.
The only ways I know to reliably maintain backups of a Windows VM at an enterprise level with something like Proxmox are:
- Since we are talking specifically a SQL Server machine, you keep the data separated because you backup with SQL Server's own functionalities. The OS you can use whatever, or just keep a spare/clone SQL Server installation (no license violation if it's stopped). However, you are using SQL Server Express, which is troublesome because you cannot have scheduled tasks, so you need to find something else.
- Using Windows (Server) Backup from inside the VM, maybe coupled with a ZFS shared folder that you snapshot after the backup is taken, so that you can actually control the history you want to keep from the storage management side; this is free and actually works pretty well; recovery is not the fastest, as you need to boot the Windows installation iso and perform a fake "bare metal" restore inside the VM. You need to start the SQL Server VSS writer, and that should give you consistent backups of SQL Server, too, in theory. I do this regularly on development and test environments, but production is a different beast, and documentation is not the clearest (thanks Microsoft, good job as usual).
- Use other certified backup solutions that do better job than Windows Backup, but again from inside the VM; still costly if you have to buy licenses for a lot of VMs, so at this point I would consider going for the big guys.
In fact, I know you are asking about Proxmox, but it must be said that professionally, the best way to manage Windows machines that bear important data is using VMWare vSphere or Hyper-V with Veeam Backup; it is the most costly solution, but also the coolest for disaster recovery: you can run the machine from backups, for example, and other cool features.
I am not advertising anything; I have experienced pretty much every virtualization platform that makes sense, and if you go Windows, that's how it is.