You should never need a second gateway. Putting in a second gateway can even cause more issues.
A gateway is only to tell your system where to go when an IP is NOT in any of the local IP-ranges that it knows about.
VLAN can help to seperate networks, but each vlan still need a seperate subnet to route properly between the different VLAN's
If you want the backup-process to run through this interface, you will need to put a second IP / IP-range on the target and put that same one on your source.
If that is not possible, the only other option would be use a smaller subnet within your main subnet, but that's more advanced and will be less clear/flexible.
For example, if your main range is 192.168.0.X/24, you could use the IP's 192.168.0.9 to 192.168.0.14 for ALL the devices (so Proxmox and the backup-servers/-storages) that you want to communicate for backup. On Proxmox you would then use subnet /29, so it uses that to communicate to other things, and on the storage and the like you use /24 still to reach the router and the like.