The computer backup systems do tend to be amp limited. They are designed to keep system up for the few minutes it takes to shut them down gracefully, not to keep them running for long periods of time.
Doing your own makes it easy to size the battery to run pretty much anything you want for as long as you need, but designing the electronics to monitor current power, monitor current batter charge, switch instantly when power drops, and monitor battery power to make sure you do not destroy the batteries by dropping their charge too low is not an easy task.
The type of batteries you use can also a problem spot. Normal car batteries are cheap, but they do not do well with discharges. They are designed to give one powerful starting discharge, and then be recharged immediately. You need the more expensive multi-discharge deep-discharge batteries, often listed as marine batteries.
Most people that want that kind of long running power just use how ever many batteries they decide they need hooked together on a trickle charger. Then they manually attach the batteries to the system when needed.
Good luck, and if you manage to DIY a truly functional system to act as a battery backup then post up the specs.