Estimate the price of a couple options, and have the organization - whomever pays the bills - help decide. Both downtime and minimizing downtime have costs. Include costs of downtime and your time, as well as anything purchased.
Backup restore is using tech you already have, so possibly no software cost. You should already be taking backups and doing the occasional restore test, and so have an idea of how long this takes. If you have never done a restore, this is an excellent reason to do one. A simple shut it down and restore is easy to do correctly maintaining data integrity, but could possibly incur hours of downtime. Recovery time and recovery point likely could be improved by being more clever with the restore.
Minimalizing downtime of the system requires some design and infrastructure. Can the applications be made highly available, by being load balanced across multiple compute VMs? Then an option could be to bring up a new instance, have both behind a load balancer for some time, then drain and remove the old instance. High cost of infrastructure, design, and maintenance time, but small downtime of the service. Further, maintaining separate compute instances can be useful, for example during OS upgrades.
And the live migration option. Almost certainly vMotion Storage will involve paying VMware, have sales tell you how much. Also live storage migration has considerable network demands. What this investment buys is a command to move a VM with almost zero downtime. Although this does not exercise your restores, it does not count as backup in a business continuity plan.