I use a UPS in each of my computers, with enough battery power to keep them running perhaps 10 minutes. Each UPS came with software that can sense the battery charge running low, and at least when running Windows, tell the computer to go into sleep mode.

This apparently starts a high priority task that suspends all other tasks, then writes the contents of all RAM not used by that task to a special file on the hard drive, and modifies the system boot code to offer a wakeup mode. The high priority task then shuts down the computer.

The wakeup mode of booting does not go through much of the usual startup procedure. Instead, it will reload most of memory (including the operating system) from this special file, then allow all other tasks to resume from the point of interruption.

You might want to check if any brands of UPS offer similar software for the operating system on your servers.

Note that the server software may need to detect that this has happened, and if so, modify the way it handles any timeouts that happened during sleep mode.

Also, beware of any model of UPS where a VA rating is specified but a wattage rating is not; assuming that those two ratings are equal is very likely to result on buying an inadequate UPS.