Have you tried the power interrupt with a very light load, like
a 4W night light? That might give you a clue whether the system
will run with a new battery.
Use a voltmeter to measure the voltage on the UPS battery when
unloaded. Turn on the system, do the power interruption and see what
the battery voltage does. Try it with a very light load.
If you're paranoid, put a 20 amp fuse between the car battery and
ups battery.
Another option is to hook a car headlight or some other heavy load
directly to the UPS battery to see if it is good.
Been my experience that a new UPS on sale is about the same cost
as a quality replacement battery for your old one.
I don't have anything mission critical. If my UPS runs the system
for 10 seconds, it's done its job. Any power outage longer than that
will likely be out for way longer than the run time of any battery
system I'm willing to devote to the system.