My UPS has a 5ms switchover time, server shuts off during said time
Typical PC power supply hold-up time is about 20ms, according to PC Power supply specifications and certifications web site. It means that your server should be able to survive 5ms switchover without any problems.
So. One of the following:
Either your server's power supply's capacitors are near the end of their life span, so it cannot hold even for 5ms, in which case I'd say you need to replace the power supply since it's not within any reasonable spec.
Or your UPS switchover time is considerably larger than 5ms, in which case you need to get another UPS since this one is faulty (not underpowered, just bad, warranty case, etc.)
Your server requires more than 1500 VA your UPS can supply, in which case UPS shuts down because it's overloaded.
First, rule out #3 above. Then, the easiest way to know if it's #1 or #2 is to try a different server with that UPS, or a different UPS with your server.
The server shuts off because it's out of power. If the Eaton UPS cannot get power to the server withint the appropriate amount of time to prevent the server running out of power, the UPS is simply not capable and you should go with your instincts and purchase a better UPS.
You need an 'on-line' UPS. That means that the UPS is ALWAYS providing power and simultaneously keeping it's battery topped up. When incoming power fails, it just keeps right on providing power. With these guys there is NO switch-over time during which your server isn't getting power.
You should only use something with a switch over time if you can prove that your server is able to survive that time (as in, you can look at the server's specs and see that it has X milliseconds of survival, and X is greater than the 5 ms your UPS takes to switch).
I think most of APC's products fall into the 'on-line' category, but be sure to examine the specs carefully before buying.