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It makes the UPS switch from outlet power to battery power just for a moment. I'll also note that the lights sort of flicker"
Not surprising the room lights sometimes flicker. Some ATX PSUs momentarily pull a very high current out of the wall socket (and through your UPS) at switch on.
Some PSUs are worse than others. I couldn't find current surge figures for your 1300W unit, but here is a cheap 550W PSU that pulls a hefty 86.31A from a 230V AC input (my mains is 240V). The switch on surge is usually lower for 115V AC supplies, but can still be quite large. With any luck, your Asrock PSU will have a better designed inrush current limiter.
https://www.tomshardware.com/reviews/xpg-pylon-550w-power-supply-review/2
Inrush current, or switch-on surge, refers to the maximum, instantaneous input current drawn by an electrical device when it is first turned on. A large enough inrush current can cause circuit breakers and fuses to trip. It can also damage switches, relays, and bridge rectifiers. As a result, the lower the inrush current a PSU has right as you turn it on, the better.
It's good to see your APC UPS is rated at 1500VA/900W. I doubt you'll ever exceed 900W unless you hang several more computers on the system. However, you do have a "brute" of a PSU and it obviously affects the operation of your UPS.
I have a similar APC UPS also rated at 1500VA, but it's physically bigger than your unit and probably a different design. The link below is for the 120V version, but mine is the 230V version. I think the main difference is mine holds two 12V batteries, not one.
https://www.amazon.com/APC-SmartConnect-Interactive-Uninterruptible-SMC1500C/dp/B077Y62GSJ
I had three PCs and monitors running on it yesterday, plus two hardware firewalls and a bunch of network switches. None of them were consuming huge amounts of power, but I don't see any flickering lights when switching on systems.
It may just be that your 1300W PSU would be happier on a 2200VA or 3000VA UPS as regards switch on surge. I'm not suggesting you should go out and buy a 2200W UPS though.
https://www.amazon.com/APC-Smart-UPS-SmartConnect-Uninterruptible-SMT2200C/dp/B077YDWR3Y
Just be aware that high switch on currents can eventually burn out electrical contacts in mains switches and mains relays. I've replaced various switched mains outlet sockets and relays over the years (in series with ATX PSUs) and discovered melted and blackened contacts inside. It can take years before things fail but cheap switches/relays go sooner than quality products.