Regardless of VM or LXC, I would only install docker once. There's generally no need to create multiple docker VMs/LXCs on the same host. Unless you have a specific reason; like isolating outside traffic by creating a docker setup for only public services.
Backups are the same with VM or LXC on Proxmox.
The main advantages of LXC that I can think of:
- Slightly less resource overhead, but not much (debian minimal or alpine VM is pretty lightweight already).
- Ability to pass-through directories from the host.
- Ability to pass-through hardware acceleration from a GPU, without passing through the entire GPU.
- Ability to change CPU cores or RAM while it's running.