By default, the Node management LIF, which lives on e0M, which can't do VLANs, is the LIF used for sending ASUP.
Just adding the route will only work if it's done properly, and if your network enables outbound NAT/PAT for the IPs of the storage. It sounds like they don't, so no, this wouldn't work.
Easiest option is to setup the NetApp system to send ASUP by a proxy server on your network, or change transport method to SMTP.
However, if you are in a situation where you can test, last time I tried, you could create LIFs of type node-mgmt on a different VLAN (and therefore not on e0M, and therefore routable) for each node. However, this is a non-standard configuration, and may break during upgrades.